Ion trap

ABSTRACT

An ion trap comprising: a first array of magnetic elements arranged to generate a first magnetic field with a degree of homogeneity; and an array of electrodes arranged to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field has a substantially maximum degree of homogeneity; wherein the array of electrodes is planar and parallel to the direction of the magnetic field at the location; and wherein a primary first magnetic element is arranged to generate a first component of the first magnetic field and other first magnetic elements are arranged to generate compensating components of the first magnetic field that reduce the gradient, the curvature and higher order derivatives of the first component of the first magnetic field at the location where the first magnetic field has the substantially maximum degree of homogeneity.

FIELD OF THE DISCLOSURE

The disclosure relates to an ion trap, and to a mass spectrometer and acomponent of a superconducting microwave quantum circuit incorporatingthe ion trap.

BACKGROUND TO THE DISCLOSURE

An ion trap is an apparatus used to confine or isolate a chargedparticle, such as an electron. One class of such apparatus is known asPenning traps. In general a Penning trap uses a magnetic field and anelectrostatic field together to trap charged particles. The magneticfield causes the charged particles to perform a rotational movement withthe direction of the magnetic field being the axis of the rotation. Thiseffectively confines the particles to a plane normal to the direction ofthe magnetic field. The electrostatic field is arranged to confine thecharged particles at a location along the direction of the magneticfield, by providing a potential well for the particle at the desiredlocation.

In order for a Penning trap to confine a charged particle effectivelyand to be useful for performing measurements on the trapped particle, itis important both for the magnetic field to be spatially homogeneous andfor the electrostatic field to be hyperboloid at the location thecharged particle is to be trapped. This places constraints in the designof a conventional Penning trap and its variants, generally making themcomplex and expensive.

In general, the required electrostatic potential well is created by aset of metallic electrodes with appropriate static voltages applied tothem. Conventional Penning traps are fabricated with three-dimensional(3D) electrodes. The first Penning trap used electrodes with the shapesof hyperboloids of revolution. This guarantees that the electrostaticpotential well nearly follows the ideal shape of a harmonic potentialwell. Penning traps with the electrodes of that shape are called“hyperbolic Penning traps”. In 1983 a Penning trap with electrodes withthe shapes of cylinders was introduced. This is now a common type ofconventional Penning trap and it is known as the “cylindrical Penningtrap”.

Conventional Penning traps employ a solenoid, usually superconducting,to create the required magnetic field. Solenoids are big, unscalable andvery expensive structures. In order to achieve the required spatialhomogeneity solenoid systems include, besides the main coil, additionalshim-coils with carefully chosen shim-currents. The fields created bythe shim-coils cancel inhomogeneities of the bulk magnetic field in abig volume enclosing the position where the charged particles aretrapped. The time stability of the magnetic field is achieved withpassive coils. These damp any fluctuations of the field caused byexternal magnetic noise of whatever origin. Normal conducting solenoidsare too unstable for high precision mass spectrometry and for quantumcomputation applications with trapped electrons. Superconductingsolenoids are typically room-size devices. The magnetic field in thetrapping region, i.e. the region in the immediate vicinity of thetrapped charged particles, cannot be isolated within a superconductingshield-box. The latter is the most effective protection against externalmagnetic noise. The temperature of the room-size superconductingsolenoid system cannot be regulated with a stability and accuracy belowthe level of 1 K. This is due to the big size of the solenoid, whichrequires stabilizing the temperature of the room where the solenoid withthe ion trap is located.

SUMMARY OF THE DISCLOSURE

According to a first aspect of the disclosure, there is provided an iontrap comprising:

a first array of magnetic elements arranged to generate a first magneticfield with a degree of homogeneity; and

an array of electrodes arranged to generate an electrostatic fieldincluding a turning point in electrical potential at a location wherethe magnetic field has a substantially maximum degree of homogeneity;

wherein the array of electrodes is planar and parallel to the directionof the magnetic field at the location; and

wherein a primary first magnetic element is arranged to generate a firstcomponent of the first magnetic field and other first magnetic elementsare arranged to generate compensating components of the first magneticfield that reduce the gradient and curvature of the first component ofthe first magnetic field at the location where the first magnetic fieldhas the substantially maximum degree of homogeneity.

By arranging the array of electrodes to be planar, the electrodes can befabricated much more easily than in a conventional Penning trap in whichthe electrodes are arranged as cylinders or hyperboloids of revolution.Furthermore, by arranging the array of electrodes parallel to thedirection of the magnetic field, the turning point in the electricalpotential can be made symmetric with respect to the magnetic field axis,thereby maintaining the harmonicity of the electrostatic potential welland therefore the usefulness of the trap.

One main magnetic element, which may be centrally positioned within themagnetic element array, may provide the bulk magnetic field at thetrapping position above the main electrode, which may be centrallypositioned in the electrode array. Additional magnetic elements may bein pairs and hence are below termed “shim-pairs”. Each magnetic elementof a shim-pair may be placed symmetrically at each side of the maincentral magnetic element. The shim elements ensure that the magneticfield is sufficiently homogeneous at the desired location, as thedifferent magnetic elements in the row compensate for theinhomogeneities of the magnetic field. It is also possible for theshim-pairs to be in a different plane or planes to the central magnet.

The elimination of magnetic inhomogeneities may be optimizedempirically, i.e. the shimming magnetic fields may be adjusted until thedesired degree of overall magnetic field homogeneity is achieved at thetrapping position above the central electrode in the row. The appliedcurrents or the applied magnetisation of the shim-pairs can always bechosen to eliminate the magnetic inhomogeneities up to some degree,which depends on the number of employed shim-pairs. The shimming processmay ensure that the magnetic field is sufficiently homogeneous at thetrapping position above the central electrode in the middle of the row.The magnetic sensor used for probing and optimising the magnetic fieldat the trapping region may be the trapped particle itself, mostconveniently, but not exclusively, an electron. This is the one of themost sensitive magnetic sensors available in nature.

The magnetic element array may comprise a row of magnetic elements,which row extends in the same direction as the row of electrodes. Themagnetic elements may be either wires made of normal conducting orsuperconducting materials or permanent magnets fabricated withconventional ferromagnets or permanently magnetised superconductors.

It is possible for the electrodes of the array to be at differentheights within the array or to have contoured surfaces. However, it ispreferred that the electrodes of the array each have surfaces facing thelocation where the magnetic field is substantially homogeneous and thatthese surfaces are substantially coplanar.

The array of electrodes may comprise a row of three or more electrodes,which row is arranged to be parallel to the direction of the magneticfield at the location where the magnetic field is substantiallyhomogeneous. Typically, the row comprises five electrodes.

Usually, the lengths of the electrodes along the direction of the roware such that an electrode in the middle of the row is shortest andelectrodes at the ends of the row are longest. This facilitates theelectrostatic equipotential lines to become hyperbolic along the lengthof the row (that is also the direction of the magnetic field), above thecentral electrode in the row.

The electrode array may comprise a guard electrode on each side of therow. Typically, the guard electrodes are coupled to ground, but it isalso possible for an electric potential to be applied to the guardelectrodes. The guard electrodes ensure that the electric field has awell defined turning point above the electrode in the middle of the rowin a direction across the width of the row. Moreover, the guardelectrodes help to shape the electric field such that the trapped ionfinds an equilibrium position in the electric field at a non-zero heightabove the electrode in the middle of the row.

The guard electrodes can overlap the electrodes of the row, withoutintersecting them. This can reduce the effect of the insulating gapsbetween any neighbour electrodes on the shape of the electric filed.

Helpfully, the array of electrodes may be provided on a substrate andthe magnetic field generator is provided on the same substrate. Indeed,according to a second aspect of the disclosure there is provided an iontrap comprising:

a substrate;

a magnetic field generator provided on the substrate and arranged togenerate a magnetic field; and

an array of electrodes provided on the same substrate and arranged togenerate an electrostatic field including a turning point in electricalpotential at a location where the magnetic field is substantiallyhomogeneous,

wherein the array of electrodes is planar and parallel to the directionof the magnetic field at the location.

This significantly simplifies fabrication of the ion trap, by allowingthe magnetic field generator to be provided close to the electrodearray. This eliminates the need for expensive superconducting coils togenerate a sufficiently strong magnetic field from a remote location, asin conventional Penning traps, as the magnetic field close to themagnetic field generator can be utilised. It can also mean that the iontrap can be manufactured as a small scale integrated electronic device,for example using thin or thick film fabrication techniques.

The electrode array may be provided on a top surface of the substrateand the magnetic field generator provided below the electrode array.

The ion trap has a number of useful applications. In particular, thereis provided a mass spectrometer comprising the ion trap, and there isprovided a building block for microwave quantum circuits.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of an ion trap according to a firstpreferred embodiment.

FIG. 2 is an exploded schematic illustration of the ion trap omitting asubstrate.

FIG. 3 is a graphical illustration of electrostatic potential along thelength of a row of electrodes of the ion trap.

FIG. 4 is a graphical illustration of electrostatic potential across thewidth of the row of electrodes on the ion trap. That is in the directionparallel to the normal of the plane where the electrodes are sitting. Itis the direction along the “y” axes, as defined in FIG. 1.

FIG. 5 is a graphical illustration of an optimal tuning ratio for theion trap.

FIG. 6 is a graphical illustration of a magnetron ellipse of the iontrap.

FIG. 7 is a graphical illustration of an optimal tuning ratio of the iontrap as a function of the height y₀. The height y₀ is the equilibriumposition of the trapped ions, above the central electrode in the row.

FIG. 8 is a graphical illustration of variation of the height y₀ as afunction of the applied voltage ratios T_(c), T_(e). This graphicassumes a row with 5 electrodes.

FIG. 9 is a graphical illustration of vertical anharmonicity C₀₁₂ alonga compensated path of the ion trap.

FIG. 10 is a graphical illustration of quadratic axial frequency shiftsproduced by the anharmonicities C₀₀₆, C₀₁₄.

FIG. 11 is a graphical illustration of the cubic axial frequency shiftproduced by the anharmonicity C₀₀₈.

FIG. 12 is a graphical illustration of the optimal trapping positions y₀⁰¹² and y₀ ⁰¹⁶ as a function of a length l_(c) of compensationelectrodes of the ion trap.

FIG. 13 is a graphical illustration of an optimal tuning ratio of theion trap as a function of the length l_(c) of compensation electrodes.

FIG. 14 is a graphical illustration of axial dip of a single electrontrapped in the ion trap.

FIG. 15 is another graphical illustration of axial dip of a singleelectron trapped in the ion trap.

FIG. 16 is a schematic illustration of a cryogenic mass spectrometerincorporating the ion trap.

FIG. 17 is a schematic illustration of a waveguide incorporating the iontrap.

FIG. 18 is a schematic illustration of the ion trap coupled to anothermicrowave circuit.

FIG. 19 is a schematic illustration of the magnetic elements, in thiscase with the shim-pairs placed above the main magnetic element.

FIG. 20 is a calculated example of the z-component, B_(z), of themagnetic field along the axes û_(z), with and without magneticcompensation of inhomogeneities.

FIG. 21 is a calculated example of the y-component of the magnetic fieldalong the axes û_(z), with and without magnetic compensation. Theundesired vertical component of the magnetic field, B_(y), is shown tovanish with the compensation.

FIG. 22 zooms the same calculated example as in FIG. 20 in a smallerarea around the interesting trapping position (0, y₀, 0).

FIG. 23 is a calculated example of the z-component, B_(z), of themagnetic field along the vertical axes û_(y), with and without magneticcompensation of inhomogeneities. It shows how the homogeneity of B_(z)is achieved.

FIG. 24 is a schematic illustration of the magnetic elements fabricatedwith closed-loop wires and enclosed in a superconducting shielding case.

FIG. 25 is a schematic illustration of the currents enclosed in thesuperconducting shielding case.

FIG. 26 is a sketch of the magnetic elements fabricated with hightemperature superconductors. The graph shows the corresponding magneticdipole densities.

FIG. 27 is a schematic illustration of the magnetic elements forcreating the trapping magnetic field and the magnetic elements foreliminating the earth's magnetic field along the û_(x) axes, B_(x)^(earh).

FIG. 28 is a schematic illustration of the magnetic elements forcreating the trapping magnetic field and the magnetic elements foreliminating the earth's magnetic field along the û_(y) axes, B_(y)^(earh).

FIG. 29 is a schematic illustration of the magnetic elements forcreating the trapping magnetic field and the magnetic elements foreliminating the earth's magnetic field along the û_(x) axes, B_(x)^(earh) and along the û_(y) axes, B_(y) ^(earh).

FIG. 30 is a schematic illustration of the complete ion chip, includingthe electrodes for creating the trapping electrostatic potential, themagnetic elements for creating the trapping magnetic field and themagnetic elements for eliminating the earth's magnetic field along theû_(x) axes, B_(x) ^(earth) and along the û_(y) axes, B_(y) ^(earh).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIGS. 1 and 2, an ion trap 1 comprises an array 2 ofmagnetic elements and an electrode array 3 provided on a substrate 4.The ion trap 1 is effectively a flat variant of a conventional Penningtrap, and can be referred to as a Penning trap in the context of thegeneral principles of its operation, although not in terms of itsspecific structure, which differs significantly from conventionalthree-dimensional traps. The term “coplanar waveguide Penning trap” hasbeen coined to describe the ion trap 1.

It should be noted that the ion trap 1 is described in the context of atrap for negatively charged ions, in particular an electron, as this islikely to be the most common use of the ion trap 1 for applications as abuilding block of microwave quantum circuits. However, the skilledperson will recognise that the trap can equally be used for trappingpositively charged ions by reversing the polarity of the electrode array3.

In this embodiment, the substrate 4 is a dielectric material suitablefor use as a waveguide for microwaves, for example in the range 1 to 30GHz. Suitable materials are sapphire or quartz or other insulators withlow dielectric losses.

The electrode array 3 comprises a ring electrode 6, two compensationelectrodes 7, 8 and two end cap electrodes 9, 10 arranged in a row 5.The ring electrode 6 is in the middle of the row 5 and the end capelectrodes 9, 10 are at the ends of the row 5. The compensationelectrodes 7, 8 are each between a respective end cap electrode 9, 10and the ring electrode 6. In other embodiments, the compensationelectrodes 9, 10 are omitted and the row 5 includes only the ringelectrode 6 and end cap electrodes 9, 10. On each side of the row 5 is aguard electrode 11, 12. These guard electrodes are also known as “groundplanes” when the trap is viewed simply as a section of a coplanarwaveguide. The coplanar waveguide is a flat transmission line widelyused in microwave applications.

Wires (not shown) are provided so that electric potential can be appliedto the electrodes 6, 7, 8, 9, 10, 11, 12. In this embodiment, the twocompensation electrodes 7, 8 are electrically coupled to one another bythe wires, the two end cap electrodes 9, 10 are electrically coupled toone another by the wires and the two guard electrodes 11, 12 areelectrically coupled to one another by the wires so that the respectiveelectric potentials V_(c), V_(e), V_(g) can be applied to the pairs ofcompensation electrodes 7, 8, end cap electrodes 9, 10 and guardelectrodes 11, 12.

The electrodes 6, 7, 8, 9, 10, 11, 12 can be made of any conductingmaterial, such as gold or copper. Alternatively, they may be of a lowtemperature superconductor. Typically a conducting material issufficient when the ion trap 1 is used as a mass spectrometer, whereas asuperconducting material is appropriate when the ion trap is used inmicrowave quantum circuits (circuit-QED) applications.

The electrode array 3 has a length l_(z) in the direction of the row 5of electrodes 6, 7, 8, 9, 10, 11, 12 and a width a₀. Within this overalllength l_(z), the ring electrode 6 has a length l_(r) in the directionof the row 5, the compensation electrodes 7, 8 each have length l_(c) inthe direction of the row 5 and the end cap electrodes 9, 10 each havelength l_(e) in the direction of the row 5. Typically, the length l_(r)of the ring electrode 6 is greater than or equal to 0.1 mm, the lengthl_(c) of the compensation electrodes 7, 8 is less than or equal to 2 mmand the length of the end cap electrodes between 0.5 mm and 10 mminclusive. The lengths of the electrodes and the width are mostconveniently chosen such that they allow for the existence of a tuningratio as in FIG. 7. This figure shows a useful interval of trappingheights y₀ ∈[y₀ ^(minimum), y₀ ^(maximum)] where an optimal tuning ratioexists T_(c) ^(opt), where T_(c) ^(opt)=V_(c)/V_(r) is the ratio of thevoltage applied to the compensation electrodes to the voltage applied tothe ring. With the optimal tuning ratio T_(c) ^(opt) the first andsecond order deviations of the trapping potential well form the idealharmonic (parabolic) shape are eliminated. This guarantees that thetrapped particle very closely behaves as an ideal harmonic oscillator,with a well-defined oscillation frequency. The dimensions of theelectrodes of the ion trap are chosen such that T_(c) ^(opt) existswithin a particular, desired useful trapping interval [y₀ ^(minimum), y₀^(maximum)]. The interval can be defined arbitrarily by the user anddepends on the particular application envisaged. It is not possible togive analytical mathematical formulas to obtain the lengths and width ofthe electrodes when one particular useful interval has been chosen.Those dimensions must be obtained numerically. Different solutions mayexist. For mass spectrometry and microwave quantum circuits applicationsthe optimal solution is that which allows to cancel simultaneously allelectrostatic anharmonicities up the sixth order C₀₁₂=C₀₀₄=C₀₀₆=0. Theprocedure is as follows: once the dimensions of the trap electrodes havebeen found for a particular useful trapping interval [y₀ ^(minimum), y₀^(maximum)], the length of the correction electrodes l_(c) can beslightly varied until the optimal solution is found. This is describedin the example of FIGS. 12 and 13. The width of the electrodes and/orthe material of the substrate can also be optimized such that the inputimpedance of the ion trap is 50 Ohm, as usually required in microwavequantum circuits.

The electrodes 6, 7, 8, 9, 10, 11, 12 have lengths l_(r), l_(c), l_(e)and are arranged in the row 5 such that the row 5 is symmetrical aboutan imaginary line that bisects its length. In other words, the row 5 issymmetrical lengthwise about a centre line of the ring electrode 6.

The electrode array 3 is planar, in the sense that the electrodes 6, 7,8, 9, 10, 11, 12 are all arranged side by side, facing in the samedirection. In this embodiment, the electrodes 6, 7, 8, 9, 10, 11, 12 areprovided on a top surface of the substrate 4 and themselves each havetop surfaces that are in the same plane. In other words, the topsurfaces of the electrodes 6, 7, 8, 9, 10, 11, 12 are coplanar. However,in other embodiments, the top surfaces of the electrodes 6, 7, 8, 9, 10,11, 12 are at different heights or are contoured, whilst the electrodearray 3 still remains planar overall.

The magnetic element array 2 comprises a row 13 of a primary magneticelement 14 and four shim magnetic elements 15, 16, 17, 18. The primarymagnetic element 14 is in the middle of the row 13 and the shim magneticelements 15, 16, 17, 18 are positioned symmetrically on each side of theprimary magnetic element 14. The primary magnetic element 14 has alength l_(p) in the direction of the row 13, the shim magnetic elements15, 16 adjacent the primary magnetic element 14 each have length l_(s1)in the direction of the row 13 and the shim magnetic elements 17, 18 atthe ends of the row 13 each have length l_(s2) in the direction of therow 13. The primary magnetic element 14 is spaced apart from each of theshim magnetic elements 15, 16 adjacent the primary magnetic element 14by gaps having length l_(g1) and the shim magnetic elements 15, 16adjacent the primary magnetic element 14 are spaced apart from the shimmagnetic elements 17, 18 at the ends of the row 13 by gaps having lengthl_(g2). The length of the primary magnetic element 14 denoted by l_(p),is typically, but not necessarily, of the order of l_(p)≧l_(r)+2l_(c)(see FIG. 1). Such values of l_(p) help improve the homogeneity of themagnetic field in the trapping region. The cross section of the primarymagnetic element 14 is l_(p)×w_(p), where the thickness of the magneticelement 14 is w_(p), with values for w_(p) above 0.01 mm and below 2 cm.The choice depends on the type of material employed and the desiredmaximum value for the strength of the magnetic field at the positionwhere the charged particles are trapped.

The first shim-pair comprises the magnetic elements 15 and 16. Both areidentical in dimensions and placed symmetrically at both sides of themagnetic element 14. In FIGS. 1 and 2 the magnetic elements 15 and 16have the same thickness as the element 14: w_(p). In general, thethickness of 15 and 16 can be chosen different to w_(p). If chosendifferent, then, most conveniently, that thickness will be grater thanw_(p). This is so in order to guarantee that the current ormagnetisation of the magnetic elements 15 and 16 does not overcome thecritical values for current density and/or critical magnetic field ofthe superconducting materials employed for their fabrication. Iffabricated with not superconducting materials, the same argumentapplies, however instead of parameters such as the critical currentdensity and/or critical magnetic field the maximum value of currentdensity or maximum polarisation of the employed materials are therelevant physical constraints. The length of the magnetic elements 15and 16 is l_(s1) (see FIG. 1). There is no general constraint on thevalue of l_(s1). It may be convenient to choose l_(s1) smaller than thelength of the magnetic element 14 l_(p), in order to have the totallength of the total magnetic array (see FIG. 2) as small as possible.This choice may be convenient when more shim-pairs, such as the magneticelements 17 and 18, are implemented. Typically l_(s1)≦1 cm andl_(s1)≧0.01 mm. The separation of the magnetic elements 15 and 16 to theelement 14 is denoted l_(g1). Typically 0.001 mm≦l_(g1)≦1 cm.

In this embodiment, the magnetic elements 14, 15, 16, 17, 18 are eachsuperconducting wires coupled so as to convey an electric currentperpendicular to the length of the row 13 and parallel to the plane ofthe electrode array 3. In other embodiments, the magnetic elements 14,15, 16, 17, 18 are each high temperature superconducting magnets. Ineither case, the magnetic elements 14, 15, 16, 17, 18 are arranged togenerate a magnetic field that has a direction substantially parallel tothe row 5 of electrodes 6, 7, 8, 9, 10 and that is substantiallyhomogeneous at a location above the ring electrode 6.

Ignoring the outer edges of the electrode array 3 and instead assumingthat the outer edges extend to infinity, for simplicity, the electricfield generated by the electrode array 3 can be expressed as:

φ(x,y,z)=V _(r)·ƒ_(r)(x,y,z)+V _(c)·ƒ_(c)(x,y,z)+V_(e)·ƒ_(e)(x,y,z))+ƒ_(gaps)(x,y,z|V _(r) ,V _(c) ,V _(e))  (1)

where V_(r), V_(c) and V_(e) represent DC voltages applied to the ringelectrode 6, the compensation electrodes 7, 8 and the end cap electrodes9, 10, respectively. The functions ƒ_(r), ƒ_(c) and ƒ_(e) represent thecontribution to the electrostatic field of the ring electrode 6, thecompensation electrodes 7, 8 and the end cap electrodes 9, 10respectively. These functions ƒ_(r), ƒ_(c) and ƒ_(e) depend only on thedimensions of the electrodes 6, 7, 8, 9, 10. The function ƒ_(gaps)represents the contribution to the electrostatic field of the gapsbetween the electrodes 6, 7, 8, 9, 10 and depends on both the dimensionsof the gaps and the DC voltages V_(r), V_(c) and V_(e) applied to theelectrodes 6, 7, 8, 9, 10.

The basic functioning of the ion trap 1 can be illustrated by computingan example using equation (1). For this purpose, we choose l_(r)=0.9 mm,l_(c)=2.0 mm, l_(e)=5.0 mm, η=0.1 mm and width S₀=7.0 mm and thevoltages applied to the electrodes 6, 7, 8, 9, 10, 11, 12 are V_(r)=−1V, V_(c)=−1.15 V, V_(e)=−4 V and V_(g)=0 V. These voltages allow forcapturing electrons or any negatively charged particles around aposition directly above the centre of the ring electrode 6, which isdefined in a Cartesian reference frame at x=0, y=y₀, z=0. It isnoteworthy that the end cap electrodes 9, 10 are not grounded. Therelationship between the voltages applied to the electrodes 6, 7, 8, 9,10 in the row 5 can generally be defined as:

|V _(e) |>|V _(c) |≧|V _(r)|  (2)

in order that there is an equilibrium position at a distance y₀>0 abovethe surface of the ring electrode 6.

Referring to FIG. 3, the electric potential at distance y₀ (˜1.19 mm inthis example) above the ring electrode 6 varies in the direction z alongthe length of the row 5 of electrodes 6, 7, 8, 9, 10, having a maxima 19above the centre of the ring electrode 6 and minima 20 on each side ofthe ring electrode 6. In FIG. 4, the variation of φ along the verticalaxes y is shown.

The trapping height y₀ is determined by the equality

$\frac{\partial{\varphi \left( {0,y,0} \right)}}{\partial y} = 0.$

If the insulating gaps are vanishingly small, η→0, then ƒ_(gaps)→0. Withthis approximation, the equation for calculating y₀ is:

$\begin{matrix}{{\frac{\partial f_{r}}{\partial y} + {T_{c} \cdot \frac{\partial f_{c}}{\partial y}} + {T_{e} \cdot \frac{\partial f_{e}}{\partial y}}} = 0} & (2)\end{matrix}$

This introduces a tuning ratio

$T_{c} = \frac{V_{c}}{V_{r}}$

and an end-cap to the ring ratio

$T_{e} = {\frac{V_{e}}{V_{r}}.}$

Equation 2 shows that the trapping height depends only on voltage ratiosT_(c), T_(e)>y₀=y₀ (T_(c), T_(e)). This formal dependence holds also forthe less restrictive case that the gap η is small “enough”, η<<l_(r),l_(c), l_(e), S₀. Equation 2 cannot be solved analytically for y₀, onlynumerical values can be obtained.

The series expansion of φ(x,y,z) around the equilibrium position (x,y₀,z), including terms up to the 4^(th) order, has the following form:

φ(x,y,z)=φ(0,y ₀,0)+C ₀₀₂ z ² +C ₂₀₀ x ² +C ₀₂₀(y−y ₀)² +C ₀₁₂ z ²(y−y₀)+C ₂₁₀ x²(y−y0+C030y−y03+C202z2x2+C022z2y−y02+C220x2y−y02+C004z4+C400x4+C040y−y04  (3)

The expansion coefficients are defined by

$C_{ijk} = {\frac{1}{{1!}{j!}{k!}} \cdot \frac{\partial^{I + j + k}{\varphi \left( {x,y,z} \right)}}{{\partial x^{i}}{\partial y^{j}}{\partial z^{k}}}}$

(evaluated at (0, y₀, 0)). The symmetry of φ(x,y,z) along û_(x) andû_(z) implies that all C_(ijk) with odd i and/or odd k vanish. TheC_(ijk) coefficients define to a great extent the performance of thetrap. They (or equivalent ones) have been studied in detail forthree-dimensional cylindrical, hyperbolic and toroidal Penning traps.Moreover, as for equation 2, if the slits between electrodes are small“enough”, then C_(ijk) scale linearly with the ring voltageC_(ijk)=V_(r)·c_(ijk), where every c_(ijk)=c_(ijk)(T_(c), T_(e)) dependsonly on T_(c) and T_(e).

Plugging the series expansion 3 into Laplace equation, ∇²φ(x,y,z)=0, thefollowing equalities can be obtained:

C ₀₀₂ +C _(O20) +C ₀₀₂=0;3C _(O30) +C ₂₁₀ +C ₀₁₂=0  (4)

6C ₄₀₀ +C ₂₂₀ +C ₂₀₂=0;6C ₀₄₀ +C ₂₂₀ +C _(O22)=0;6C ₀₀₄ +C ₂₀₂ +C_(O22)=0  (5)

In the case of a 3D hyperbolic or cylindrical trap, the coordinates xand y are undistinguishable

C₂₀₀=C₀₂₀. Thus, equation 4 (left) reduces to C₀₀₂=−2C₀₂₀. From it, thepotential of an ideal Penning trap arises

φ=C₀₀₂(z²−(x²+(y−y₀)²)/2). In the case of the coplanar waveguide Penningtrap (short CPW-trap) though, x and y are distinguishable and thecurvatures C₂₀₀ and C_(O20) are not identical: C₂₀₀≠C_(O20). Hence thegeneral form of the quadrupole potential, i.e. including terms only upto the second order, is:

$\begin{matrix}{\varphi_{quad} = {{C_{002}\left( {z^{2} - \frac{\left( {x^{2} + \left( {y - y_{0}} \right)^{2}} \right)}{2}} \right)} + {\frac{1}{2}C_{002}\varepsilon \; {\left( {x^{2} - \left( {y - y_{0}} \right)^{2}} \right).}}}} & (6)\end{matrix}$

The ellipticity parameter is given by

$\varepsilon = {\frac{C_{200} - C_{020}}{C_{002}}.}$

In general, ∈≠0, and the CPW-trap is therefore an elliptical Penningtrap.

The motion of a particle in the ideal elliptical trap of equation 6 hasbeen calculated analytically (M. Kretzschmar 2008). The reducedcyclotron ω_(p)=2π ν_(p), magnetron ω_(m)=2π ν_(m) and axial ω_(z)=2πν_(z) frequencies of the trapped particle—with charge q and mass m—are:

$\begin{matrix}{{{\omega_{p} = \sqrt{{\frac{1}{2}\left( {\omega_{c}^{2} - \omega_{z}^{2}} \right)} + {\frac{1}{2}\sqrt{\left( {{\omega_{c}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}} \right)}}}};}{\omega_{m} = \sqrt{{\frac{1}{2}\left( {\omega_{c}^{2} - \omega_{z}^{2}} \right)} - {\frac{1}{2}\sqrt{\left( {{\omega_{c}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}} \right)}}}}{{\omega_{z} = \sqrt{2C_{002}\frac{q}{m}}};}{{\omega_{c} = {\frac{q}{m}B}};}{\omega_{1} = \sqrt{\omega_{c}^{2} - {2\omega_{z}^{2}}}}} & (7)\end{matrix}$

When ∈=0 the usual expressions for the frequencies of a standard“circular” (non elliptical) Penning trap are recovered. For the exampleof FIG. 3, the ellipticity is ∈=0.41. According to equation 7, thefrequencies of a trapped electron are: ω_(p)=2π·14 GHz, ω_(z)=2π·28 MHzand ω_(m)=2π·26 kHz. A magnetic field of B=0.5 T is assumed, motivatedby the suitability of the corresponding cyclotron frequency forcircuit-QED applications.

The radial motion in an ideal elliptical trap is:

(x(t),y(t)−y ₀)=(A _(p)ζ_(p) cos(ω_(p) t)+A _(m)ζ_(m) cos(ω_(m) t),A_(p)η_(p) sin (ω_(p) t)+A _(m)η_(m) sin(ω_(m) t)).  (8)

The amplitudes are given by:

${A_{p} = {\frac{1}{\omega_{p}}\sqrt{\frac{2E_{p}}{\gamma_{p}m}}}},{{\gamma_{p} = {{1 - \frac{\omega_{z}^{2}}{2\omega_{p}^{2}}} \approx 1}};}$$A_{m} = {\sqrt{\frac{2E_{m}}{\left( {\omega_{m}^{2} - {\omega_{z}^{2}\text{/}2}} \right)m}}.}$

The coefficients ζ_(p,m) and η_(p,m) have been calculated in general foran ideal elliptical Penning trap (M Kretzschmar 2008):

$\begin{matrix}{{{\xi_{p,m} = \sqrt{\frac{\omega_{c}^{2} + {{\varepsilon \mspace{11mu} \omega_{z}^{2}} \pm \sqrt{{\omega_{c}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}}}}{2\omega_{p}\text{/}\omega_{1}\sqrt{{\omega_{c}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}}}}};}{\eta_{p,m} = \sqrt{\frac{\omega_{c}^{2} + {{\varepsilon \mspace{11mu} \omega_{z}^{2}} \pm \sqrt{{\omega_{c}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}}}}{2\omega_{p}\text{/}\omega_{1}\sqrt{{\omega_{c\;}^{2}\omega_{1}^{2}} + {ɛ^{2}\omega_{z}^{4}}}}}}} & (9)\end{matrix}$

The symbols E_(p) and E_(m) represent the cyclotron and magnetronenergies, respectively. As shown by Kretzschmar, the orbit of thereduced cyclotron motion is only slightly affected by the ellipticity:ζ_(p)≈η_(p)≈1. It very nearly follows the circular shape of conventionalPenning traps. On the contrary, the magnetron motion becomes an ellipse,where the orientation of the major and minor axes (along x or y) dependson the sign of ∈. Moreover, that motion is stable for −1<∈<1 and, at thelimit |∈|→1, it becomes very slow, ω_(m)→0. In that case, the magnetronellipse tends towards a line, with an increasingly wide major axes and avanishing minor one (if ∈→+1

ζ_(m)→∞, η_(m)→0 and vice versa for ∈→−1

ζ_(m)>0, η_(m)→∞). For values |∈|≧1 the magnetron becomes an unboundedhyperbolic motion and trapping is not possible.

The ideal trap defined by the pure quadrupole potential of equation 6 isonly valid for vanishing amplitudes of the trapped particle's motion. Inreal experiments the electric anharmonicities odd and even (equation 3)must be taken into account. These generate energy-dependentfluctuations/deviations of the frequencies ω_(p), ω_(z), ω_(m).

All anharmonicities, even and odd, up to the 4^(th) order in theexpansion of φ, 3≦i+j+k≦4, (see equation 3), produce frequency shiftswhich scale linearly with the energies of the particle. Hence, they canbe expressed in matrix form:

$\begin{matrix}{{{\begin{pmatrix}{\Delta \; v_{p}} \\{\Delta \; v_{z}} \\{\Delta \; v_{m}}\end{pmatrix} = {\begin{pmatrix}M_{1,1} & M_{1,2} & M_{1,3} \\M_{2,1} & M_{2,2} & M_{2,3} \\M_{3,1} & M_{3,2} & M_{3,3}\end{pmatrix} \cdot \begin{pmatrix}{\Delta \; E_{p}} \\{\Delta \; E_{z}} \\{\Delta \; E_{m}}\end{pmatrix}}};}{M = {\begin{pmatrix}M_{1,1} & M_{1,2} & M_{1,3} \\M_{2,1} & M_{2,2} & M_{2,3} \\M_{3,1} & M_{3,2} & M_{3,3}\end{pmatrix}\mspace{14mu} {shifts}\mspace{14mu} {matrix}}}} & (10)\end{matrix}$

Each perturbation to Φ_(quad) appearing in equation 3 delivers such afrequency-shifts matrix. In total the CPW-Penning trap requires nineM^(ijk) matrices, corresponding to each C_(ijk) perturbativeHamiltonian. The expressions for all Milk are given in Appendix B. Theoverall frequency-shifts matrix is the sum of all of them:

M=M ⁰¹² +M ²¹⁰ +M ⁰³⁰ +M ²²⁰ +M ²⁰² +M ⁰²² +M ⁰⁰⁴ +M ⁴⁰⁰ +M ⁰⁴⁰  (11)

For a single electron captured in the example trap with dimensions asgiven in page 10, with the voltages of FIG. 3 and with magnetic fieldB=0.5 T, the overall frequency-shifts matrix is:

$\begin{matrix}{M = {\begin{pmatrix}{5 \cdot 10^{- 6}} & 0.5 & {- 0.9} \\{1 \cdot 10^{- 3}} & 203 & {- 411} \\{{- 2} \cdot 10^{- 3}} & {- 0.4} & 2\end{pmatrix}\mspace{14mu} {Hz}\text{/}{K.}}} & (12)\end{matrix}$

The accurate measurement of the axial frequency is essential; in mostcases the determination of the other particle's motional frequencies (orthe spin state) depend upon it. Thus, the element M_(2,2)=Δν_(z)/ΔE_(z)is the most relevant and dangerous of all frequency shifts in M. In theexample it amounts to 203 Hz/K. Such a dependence of ν_(z) on the axialenergy—which is not constant but fluctuating with the temperature—wouldrender the detection of the electron, or in general trapped chargedparticle, almost impossible, even at cryogenic temperatures.

M_(2,2) is given by the sum of M_(2,2) ⁰⁰⁴ and M_(2,2) ⁰¹². Taking intoaccount that ν_(m)<<ν_(z)<<ν_(p), we have:

$\begin{matrix}{{{M_{2,2}^{004} = {\frac{q}{16\pi^{4}m^{2}v_{z}^{3}}C_{004}}};}{M_{2,2}^{012} = {\frac{q^{2}}{32\pi^{6}m^{3}}\frac{\eta_{m}^{2}}{v_{z}^{5}}C_{012}^{2}}}} & (13)\end{matrix}$

M_(2,2) ⁰¹² is always positive, since it is proportional to the squareof C₀₁₂, while M_(2,2) ⁰⁰⁴ can be positive or negative, depending on thesign of C₀₀₄. Hence, if an appropriate optimal tuning ratio can befound, such, that the latter matrix element cancels the former T_(c)^(opt)

M_(2,2) ⁰¹²+M_(2,2) ⁰⁰⁴=0, then the linear dependence of ν_(z) upon theaxial energy can be eliminated.

The existence of T_(c) ^(opt) cannot be universally guaranteed, howeverit turns out that this is often the case. For the example trap, it canbe seen in FIG. 5, where

$\frac{\Delta \; v_{z}}{\Delta \; E_{z}}$

is plotted as a function of the applied tuning ratio. One value, T_(c)^(opt)=1.13440, eliminates M_(2,2).

Since ν_(z)∝√{square root over (C₀₀₂)}, both frequency shifts inequation 13 equally scale with the square root of the applied voltage tothe ring electrode (the central electrode in the row 5) V_(r) ^(−1/2).The equation M_(2,2)(T_(c) ^(opt))=0 is independent of the actual valueof the potential applied to the central electrode in row 5 and is solelydefined by the voltage ratios T_(c) and T_(e). A similar argumentapplies to the mass m and charge q. Thus, T_(c) ^(opt) is a well-definedquantity, independent of V_(r) and of the trapped atomic species. Itdoes change with, T_(e) but this is simply equivalent to an inevitabledependence upon the trapping position y₀ (see FIGS. 7 and 8). Theappearance of η_(m) in M_(2,2) ⁰¹² also implies that T_(c) ^(opt)theoretically varies with the magnetic field, however, that dependenceis negligible: −2·10⁻⁶ T⁻¹ for the example.

M_(2,2) ⁰¹² is caused by the slight dependence of the trapping height onthe axial energy, y₀=y₀ (E_(z)). Indeed: for vanishing energy E_(z)=0,y₀ is the solution to the implicit equation C₀₀₁(y₀)=0. For E_(z)>0 thatequation must be modified into C₀₀₁ (y₀′)+

z²

C₀₁₂(y₀′)=0. Here,

z²

represents the time average of A_(z) ² cos(ω_(z) ^(t))². Thus, the realheight, y₀′=y₀+Δy, depends on the axial amplitude, hence, on E_(z). Δycan be estimated as follows (we assume the approximationC₀₁₂(y₀′)≅C₀₁₂(y₀):

$\begin{matrix}{{{C_{001}\left( y_{0}^{\prime} \right)} + {{\langle z^{2}\rangle}{C_{012}\left( y_{0}^{\prime} \right)}} - {C_{001}\left( y_{0} \right)}} = {{0->{{\frac{{C_{001}\left( {y_{0} + {\Delta \; y}} \right)} - {C_{001}\left( y_{0} \right)}}{\Delta \; y}\Delta \; y} + {{\langle z^{2}\rangle}{C_{012}\left( y_{0} \right)}}}} = {{{0\Delta}\; y} = {- {\quad{{\frac{1}{2}\frac{C_{012}}{C_{020}}{\langle z^{2}\rangle}} = {\left( {{A_{z}^{2}{\langle{\cos \left( {\omega_{z}t} \right)}^{2}\rangle}} = \frac{E_{z}}{m\; \omega_{z}^{2}}} \right) = {{- \frac{1}{8\mspace{11mu} \pi^{2}m\; v_{z}^{2}}}\frac{C_{012}}{C_{020}}}}}}}}}} & (14)\end{matrix}$

At y₀+Δy, the axial potential is modified with respect to y₀. Inparticular, the E_(z)=0 axial curvature, C₀₀₂(y₀), changes to C₀₀₂(y₀′)This subsequently forces the variation of a), as a function of E_(z):

$\begin{matrix}{{\Delta\omega}_{z} = {{{\frac{\partial\omega_{z}}{\partial y} \cdot \Delta}\; y} = {{\frac{q}{m}\frac{1}{\sqrt{\frac{2q\; C_{002}}{m}}}\frac{\partial C_{002}}{\partial y}\Delta \; {y\frac{\Delta \; v_{z}}{\Delta \; E_{z}}}} = {{- \frac{q^{2}}{32\pi^{6}m^{3}}}\frac{C_{012}^{2}}{v_{z}^{5}}\left( \frac{C_{002}}{2_{020}} \right)}}}} & (15)\end{matrix}$

The model described can be tested by computing numerically the radialmotion of an electron in a real CPW-Penning trap, using the potential ofequation 1 without approximations. The numerical calculation shows avertical shift of the radial ellipse relative to the ideal one. Anexample, based upon the trap of section, is plotted in FIG. 6. This hasbeen computed assuming axial energy E_(z)=4.2 K, magnetron energy

$E_{m} = {{- \frac{v_{m}}{v_{z}}}E_{z}}$

and vanishing cyclotron energy. The shift predicted by equation 14amounts to Δy=0.355 μm and is in good agreement with the numericalresult of Δy=−0.325 μm.

An optimal tuning ratio can be found within a continuous interval oftrapping heights, however it varies smoothly as a function of y₀. Thisis shown in FIG. 7, where the plot of the optimal tuning ratio, T_(c)^(opt), versus y₀ is presented. A useful interval exists (0.6 mm≦y₀≦1.3mm, for the example), where M_(2,2) can be eliminated. Beyond the upperor lower bounds of that interval, the optimal tuning ratio does notexist.

As shown in FIG. 8, T_(c) and T_(e) can be tuned independently andmultiple combinations can be found to obtain one particular trappingposition. However, the compensated interval is determined by a univocalrelationship, y₀

(T_(e), T_(c) ^(opt)), as featured in FIG. 8. It must be noted also thatT_(e) is the main parameter for changing y₀, while T_(c) is basicallyused for compensation.

After eliminating the linear dependence of the axial frequency uponE_(z), non-linear shifts may still be important, particularly when y₀ issmall. The next most significant even anharmonicities, whose effect canbe calculated by first order perturbation theory, are C₀₀₆ and C₀₀₈.These produce the following quadratic and cubic shifts, respectively:

$\begin{matrix}{{{{\Delta \; v_{z}} = {\frac{15\mspace{11mu} q}{128\mspace{11mu} \pi^{6}m^{3}}\frac{C_{006}}{v_{z}^{5}}\left( {\Delta \; E_{z}} \right)^{2}}};}{{\Delta \; v_{z}} = {\frac{140\mspace{11mu} q}{2048\mspace{11mu} \pi^{8}m^{4}}\frac{C_{008}}{v_{z}^{7}}{\left( {\Delta \; E_{z}} \right)^{3}.}}}} & (16)\end{matrix}$

For the example trap, these non-linear shifts are shown in FIGS. 10 and11. The next most significant odd anharmonicities, after those includedin equation 3, are: C₀₁₄, C₂₁₂, C₀₃₂, C₄₁₀, C₂₃₀ and C_(O50). Thecalculation of the corresponding frequency shifts, with rigorous secondorder perturbation theory, would be extremely cumbersome. Instead, weemploy the model presented in equations 14 and 15. Following thederivation of those equations we obtain the following frequency shifts:

$\begin{matrix}{\mspace{79mu} {{{\Delta \; v_{z}} = {{{- \frac{3\mspace{11mu} q}{128\mspace{11mu} \pi^{6}\; m^{3}v_{z}^{5}}} \cdot \frac{C_{012} \cdot C_{014}}{C_{020}}}\left( {\Delta \; E_{z}} \right)^{2}}}{{\Delta \; v_{z}} = {{{- \frac{q}{128\mspace{11mu} \pi^{6}m^{3}v_{z}^{3}}} \cdot \frac{C_{012} \cdot C_{212}}{C_{020}}}\Delta \; {E_{z}\left( {{\frac{\xi_{p}^{2}}{\gamma_{p}v_{p}^{2}}\Delta \; E_{p}} + {\frac{\xi_{m}^{2}}{v_{m}^{2} - {v_{z}^{2}\text{/}2}}\Delta \; E_{m}}} \right)}}}}} & (17)\end{matrix}$

The shift due to the term C₀₁₄ predicted by equation 17 has a similarmagnitude as the one produced by C₀₀₆ in equation 16. It must be takeninto account when designing a coplanar-waveguide Penning trap. Noticethat equation 17 both frequency shifts, produced by C₀₁₄ and C₂₁₂ vanishat the position where the term C₀₁₂ vanishes. This position is denotedby y₀ ⁰¹². For the trap of our example, the position y₀ ⁰¹² is shown inFIG. 9.

Equation 17 predicts an axial frequency shift due to C₂₁₂ scaling withthe products ΔE_(z)·ΔE_(p) and ΔE_(z)·ΔE_(m). In the former case, theshift is proportional to 1/ν_(p) ²; hence, it is normally negligible. Inthe latter case, the fluctuations of the magnetron energy, ΔE_(m), arevery small and the corresponding value of Δν_(z) is also negligible.Thus, C₂₁₂ can be ignored. The same arguments apply to C_(O32), whichproduces a shift very similar to that of equation 17 for C₂₁₂ (ξ_(p,m)must be simply substituted by η_(p,m)). The remaining 5^(th) ordercoefficients, C₄₁₀, C₂₃₀, C₀₅₀, generate only deviations of thecyclotron and magnetron frequencies ν_(p) and ν_(m) with products ofΔν_(p) and Δν_(m). Therefore they can be ignored too, since thosefluctuations are very small, as can be seen from the linear frequencyshifts matrix (see example of equation 12). Finally, similar argumentsapply to all 6^(th) order coefficients which have not been considered inequation 17 and the following discussion, namely C₂₂₂, C₂₀₄, C₀₂₄, C₄₂₀,C₄₀₂, C₀₄₂, C₂₄₀, C₆₀₀, C₀₆₀; they are all irrelevant.

FIG. 10 reveals the existence of one particular position, y₀ ⁰⁰⁶˜0.83mm, at which the coefficient C₀₀₆ vanishes. The question which arises iswhether the trap can be designed to make y₀ ⁰⁰⁶ and y₀ ⁰¹² coincident,thereby optimizing the compensation electrode. The answer is affirmativeand is illustrated in FIG. 12. It shows the variation of y₀ ⁰¹² and y₀⁰⁰⁶ when changing the length of the compensation electrode, l_(c), whilekeeping all other dimensions of the trap constant. For the example, whenl_(c)˜1.84 mm

y₀ ⁰¹²=y₀ ⁰⁰⁶. This result is also visible in FIG. 13, where theintersection of the two curves shows the optimised length of thecorrection electrode. For this optimized trap, C₀₀₄=C₀₁₂=C₀₀₆=0 at y₀⁰¹².

With the dependence ν_(z)=ν_(z)(E_(z), E_(z) ², E_(z) ³) given inequations 13, 16 and 17, it is possible to perform a realisticsimulation of the axial signal of a trapped electron. Assuming thedetection scheme employed in many Penning trap experiments, the signalappears as a shortcut (=the axial dip) of the resonance resistance of anexternal detection parallel LC-circuit. The goal is to compare theactual detection signal of a trapped electron in the real trap, to thedetection signal of the ideal trap; the purpose is estimating the“relative visibility” of the former. The technical details are thereforeunimportant, although they can be found in the standard literature, inarticles of Gabrielse, Dehmelt and other authors. The simulations areshown in FIGS. 14 and 15. The curves are obtained by averaging the axialdip (with ν_(z)=ν_(z)(E_(z), E_(z) ², E_(z) ³)) over a Boltzmanndistribution of the axial energy. Three different values of the axialtemperature 7′, have been analysed.

FIG. 14 shows the reduction of the “visibility” of the dips withincreasing axial temperature. A random position, y₀=1.209 mm (but withoptimized tuning ratio), has been chosen for the plot. In this case,C₀₀₆ and C₀₁₄ produce the increasing deterioration of the dip, whileC₀₀₈ is negligible (see FIGS. 10 and 11). In FIG. 15 y₀=y₀ ⁰¹²=y₀⁰⁰⁶=0.820 mm. Now, C₀₀₄=C₀₁₂=C₀₀₆=0, however, C₀₀₈ still diminishes thequality of the signal with increasing axial temperature. It can beconcluded, that the detection of a single electron at 4.2 K (or lower)should be always possible within the compensated interval of the trap.However, for increasing temperatures, the non-linear anharmonicitiesmake its observation significantly more difficult, even for relativelymodest values of T.

Referring to FIG. 16, a mass spectrometer 23 according to an embodimentof the disclosure comprises the ion trap 1 located in a cryogenic vacuumchamber 24 capable of cooling the ion trap to a temperature of 4.2 K orlower. A DC voltage source 25 is provided to supply the voltages V_(r),V_(c), V_(e) to the ring electrode 6, compensation electrodes 7, 8 andend cap electrodes 9, 10. A microwave generator 26 and functiongenerator 27 are provided for injecting microwaves into the ion trap forprobing the trapped particles, and an oscilloscope 28 and Fouriertransform analyser are provided for analysing microwaves exiting the iontrap 1. Multiple ion traps 1 can be provided in the cryogenic vacuumchamber 24, allowing the mass spectrometer 23 to analyse multipletrapped particles at the same time, under similar ambient conditions.

Referring to FIGS. 17 and 18, the ion trap 1 can provide a cavity 29 formicrowaves. The cavity is equivalent to an LC circuit, as shown in FIG.17, and can be coupled to a distant microwave cavity 30 via an externalmicrowave transmission line to form a microwave quantum circuit.

Turning to the magnetic source on a chip, the Hamiltonian of a particlewith charge q and mass m in the coplanar-waveguide Penning trap is givenby:

$\begin{matrix}{H = {\frac{\left( {\overset{->}{p} - \overset{->}{A}} \right)^{2}}{2m} + {q \cdot {\varphi \left( {x,y,z} \right)}}}} & (18)\end{matrix}$

In equation 18 {right arrow over (p)} is the canonical momentum of thetrapped particle and {right arrow over (A)} is the magnetic vectorpotential. The electrostatic potential φ(x,y,z) is given by equation 1,q, m are the charge and mass of the trapped particle, respectively. Themagnetic field is calculated as {right arrow over (B)}=∇×{right arrowover (A)}. The perfectly homogeneous magnetic field is {right arrow over(B)}=B₀û_(z). The magnetic vector potential of the perfectly homogeneousmagnetic field is given by {right arrow over (A)}₀=B₀/2(xû_(y)−yû_(x)).The ideal Hamiltonian is given by

$H_{0} = {\frac{\left( {\overset{->}{p} - {\overset{->}{A}}_{0}} \right)^{2}}{2m} + {q\; {{\varphi \left( {x,y,z} \right)}.}}}$

The total Hamiltonian is therefore the sum of the ideal Hamiltonian plusa perturbative Hamiltonian: H=H₀+ΔH. The perturbative Hamiltonian isgiven by the following expression:

$\begin{matrix}{{\Delta \; H} = {{{- \frac{q}{m}}{\overset{->}{p} \cdot \Delta}\; \overset{->}{A}} + {\frac{q^{2}}{m}{{\overset{->}{A}}_{0} \cdot \Delta}\; \overset{->}{A}} + {\frac{q^{2}}{2m}\Delta \; {\overset{->}{A} \cdot \Delta}\; \overset{->}{A}}}} & (19)\end{matrix}$

The effect of the quadratic term

$\frac{q^{2}}{2m}\Delta \; {\overset{->}{A} \cdot \Delta}\; \overset{->}{A}$

can be neglected, since, for smaller deviations from the homogeneouscase, it is much smaller than the other terms in equation 19 (see forinstance Int. J. Mass Spec. Ion Proc. 141, 101, 1995). With thisapproximation, it is possible to analyse in detail the magnetic sourceas described in FIGS. 1 and 2. The magnetic elements 14, 15, 16, 17 and18, all must be aligned along the x-axes, as defined in FIG. 1. Thelength of all these elements is considerably longer than the width ofthe electrodes in the array x. The width of the electrodes is defined bythe symbol S₀, hence the length of the magnetic elements is at least afactor 5 times S₀ or even longer. With this constraint on the length ofthe electrodes, the magnetic vector potential seen at the position ofthe trapped particles, i.e. at the height y₀ above the central electrodein the row, can be assumed to have the following general form:

$\begin{matrix}{{\overset{->}{A}(x)} = {{- \frac{\mu_{0}}{4\pi}}{\int\ {{V^{\prime}}\frac{J\left( x^{\prime} \right)}{{x - x^{\prime}}}}}}} & (20)\end{matrix}$

In equation 20 the symbol μ₀ represents the magnetic permeability. Thecurrent density running along the wires is oriented along the x-axes.Assuming it is homogeneous along the whole length of the magneticelements/wires J(x′)=J₀û_(x), the magnetic vector potential istherefore:

$\begin{matrix}{{\overset{->}{A}(x)} = {{{- \frac{\mu_{0}J_{0}}{4\pi}}{\hat{u}}_{x}{\int{{V^{\prime}}\frac{1}{{x - x^{\prime}}}}}} = {{A\left( {x,y,z} \right)}{\hat{u}}_{x}}}} & (21)\end{matrix}$

With the restriction of the magnetic elements being very long ascompared to the width of the electrodes width S₀ the magnetic vectorpotential becomes a vector with component only in the û_(x) direction.Now, choosing the coulomb Gauge, this gives ∇·{right arrow over(A)}(x)=0, hence

${\frac{\partial A}{\partial x} = {\left. 0\Rightarrow A \right. = {A\left( {y,z} \right)}}},$

the magnetic vector potential is not a function of the x coordinateA≠A(x). The components of the magnetic field are therefore:

$\begin{matrix}{{{B_{x} = {{\frac{\partial A_{y}}{\partial z} - \frac{\partial A_{z}}{\partial y}} = 0}};}{{B_{y} = {{\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}} = \frac{\partial A}{\partial z}}};}{B_{z} = {{\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y}} = {- \frac{\partial A}{\partial y}}}}} & (22)\end{matrix}$

The series expansion of the magnetic vector potential function (equation21) around the trapping position (0, y₀, 0) is given by the followingexpression:

$\begin{matrix}{{A\; \left( y_{0} \right)} + {\frac{\partial A}{\partial y}\left( {y - y_{0}} \right)} + {\frac{\partial A}{\partial z}z} + {{\frac{1}{2!} \cdot \frac{\partial^{2}A}{\partial y}}\left( {y - y_{0}} \right)^{2}} + {{\frac{1}{2!} \cdot \frac{\partial^{2}A}{\partial z^{2}}}z^{2}} + {{\frac{1}{{1!} \cdot {1!}} \cdot \frac{\partial^{2}A}{{\partial y}{\partial z}}}\left( {y - y_{0}} \right){z++}{\frac{1}{{2!} \cdot {1!}} \cdot \frac{\partial^{3}A}{{\partial y^{2}}{\partial z}}}\left( {y - y_{0}} \right)^{2}z} + {{\frac{1}{{1!} \cdot {2!}} \cdot \frac{\partial^{3}A}{{\partial y}{\partial z^{2}}}}\left( {y - y_{0}} \right)z^{2}} + {{\frac{1}{3!} \cdot \frac{\partial^{3}A}{\partial y^{3}}}\left( {y - y_{0}} \right)^{3}} + {{\frac{1}{4!} \cdot \frac{\partial^{4}A}{\partial z^{4}}}{z^{4}++}{\frac{1}{4!} \cdot \frac{\partial^{4}A}{\partial y^{4}}}\left( {y - y_{0}} \right)^{4}} + {{\frac{1}{{2!} \cdot {2!}} \cdot \frac{\partial^{4}A}{{\partial y^{2}}{\partial z^{2}}}}\left( {y - y_{0}} \right)^{2}z^{2}} + \ldots} & (23)\end{matrix}$

The symmetry of the magnetic array (see FIGS. 1 and 2) implies thatA(y,z)=A(y,−z), ∇z. Thus, all odd derivatives of A(y,z) with z vanish atthe trapping position (0, y₀,0). The series expansion of A(y,z),simplifies to:

$\begin{matrix}{{A\left( {y,z} \right)} = {{A\left( y_{0} \right)} + {\frac{\partial A}{\partial y}\left( {y - y_{0}} \right)} + {{\frac{1}{2!} \cdot \frac{\partial^{2}A}{\partial y^{2}}}\left( {y - y_{0}} \right)^{2}} + {{\frac{1}{2!} \cdot \frac{\partial^{2}A}{\partial z^{2}}}z^{2}} + {{\frac{1}{{1!}{2!}} \cdot \frac{\partial^{3}A}{{\partial y}{\partial z^{2}}}}\left( {y - y_{0}} \right)z^{2}} + {{\frac{1}{3!} \cdot \frac{\partial^{3}A}{\partial y^{3}}}\left( {y - y_{0}} \right)^{3}} + {{\frac{1}{4!} \cdot \frac{\partial^{4}A}{\partial z^{4}}}z^{4}} + {{\frac{1}{4!} \cdot \frac{\partial^{4}A}{\partial y^{4}}}\left( {y - y_{0}} \right)^{4}} + {{\frac{1}{{2!}{2!}} \cdot \frac{\partial^{4}A}{{\partial y^{2}}{\partial z^{2}}}}\left( {y - y_{0}} \right)^{2}z^{2}} + \ldots}} & (24)\end{matrix}$

The constant A(y₀) has no dynamic effect and can be ignored. Now, takinginto account the expressions for the magnetic field components B_(y),B_(z) of equation (22), the series expansion of the magnetic vectorpotential can be written as follows:

$\begin{matrix}{{A\left( {y,z} \right)} = {{- {B_{z}^{0}\left( {y - y_{0}} \right)}} - {{\frac{1}{2!} \cdot \frac{\partial B_{z}}{\partial y}}\left( {y - y_{0}} \right)^{2}} + {{\frac{1}{2!} \cdot \frac{\partial B_{y}}{\partial z}}z^{2}} - {{\frac{1}{{1!}{2!}} \cdot \frac{\partial^{2}B_{z}}{\partial z^{2}}}\left( {y - y_{0}} \right)z^{2}} - {{\frac{1}{3!} \cdot \frac{\partial^{2}B_{z}}{\partial y^{2}}}\left( {y - y_{0}} \right)^{3}} + {{\frac{1}{4!} \cdot \frac{\partial^{3}B_{y}}{\partial z^{3}}}z^{4}} - {{\frac{1}{4!} \cdot \frac{\partial^{3}B_{z}}{\partial y^{3}}}\left( {y - y_{0}} \right)^{4}} - {{\frac{1}{{2!}{2!}} \cdot \frac{\partial^{3}B_{z}}{{\partial y}{\partial z^{2}}}}\left( {y - y_{0}} \right)^{2}z^{2}} + \ldots}} & (25)\end{matrix}$

Hence, this gives the following magnetic vector potential terms:

$\mspace{79mu} {{\overset{\rightarrow}{A}}_{0} = {{- {B_{z}^{0}\left( {y - y_{0}} \right)}}{\hat{u}}_{x}\mspace{14mu} \left( {{ideal}\mspace{14mu} {part}} \right)}}$$\mspace{79mu} {{\overset{\rightarrow}{A}}_{1} = {\left( {{{{- \frac{1}{2!}} \cdot \frac{\partial B_{z}}{\partial y}}\left( {y - y_{0}} \right)^{2}} + {{\frac{1}{2!} \cdot \frac{\partial B_{y}}{\partial z}}z^{2}}} \right){\hat{u}}_{x}}}$$\mspace{79mu} {{\overset{\rightarrow}{A}}_{2} = {\left( {{{{- \frac{1}{{1!}{2!}}} \cdot \frac{\partial^{2}B_{z}}{\partial z^{2}}}\left( {y - y_{0}} \right)z^{2}} - {{\frac{1}{3!} \cdot \frac{\partial^{2}B_{z}}{\partial y^{2}}}\left( {y - y_{0}} \right)^{3}}} \right){\hat{u}}_{x}}}$${\overset{\rightarrow}{A}}_{3} = {\left( {{{\frac{1}{4!} \cdot \frac{\partial^{3}B_{y}}{\partial z^{3}}}z^{4}} - {{\frac{1}{4!} \cdot \frac{\partial^{3}B_{z}}{\partial y^{3}}}\left( {y - y_{0}} \right)^{4}} - {{\frac{1}{{2!}{2!}} \cdot \frac{\partial^{3}B_{z}}{{\partial y}{\partial z^{2}}}}\left( {y - y_{0}} \right)^{2}z^{2}}} \right){\hat{u}}_{x}}$${\overset{\rightarrow}{A}}_{4} = {\left( {{{{- \frac{1}{5!}} \cdot \frac{\partial^{4}B_{y}}{\partial y^{4}}}\left( {y - y_{0}} \right)^{5}} - {{\frac{1}{{3!}{2!}} \cdot \frac{\partial^{4}B_{z}}{{\partial y^{2}}{\partial z^{2}}}}\left( {y - y_{0}} \right)^{3}z^{2}} - {{\frac{1}{{1!}{4!}} \cdot \frac{\partial^{4}B_{z}}{\partial z^{4}}}\left( {y - y_{0}} \right)z^{4}}} \right){\hat{u}}_{x}}$

Employing Maxwell's equations, the following relationships among thederivatives of the magnetic field components are obtained:

$\begin{matrix}\begin{matrix}{{\nabla{\times \overset{\rightarrow}{B}}} = \left. \overset{\rightarrow}{0}\Rightarrow\frac{\partial B_{z}}{\partial y} \right.} \\{{= \frac{\partial B_{y}}{\partial z}};}\end{matrix} & (26) \\\begin{matrix}{{\nabla{\cdot \overset{\rightarrow}{B}}} = \left. 0\Rightarrow{\frac{\partial B_{x}}{\partial x} + \frac{\partial B_{y}}{\partial y} + \frac{\partial B_{z}}{\partial z}} \right.} \\{= \left. 0\Rightarrow{\frac{\partial B_{y}}{\partial y} + \frac{\partial B_{z}}{\partial z}} \right.} \\{= \left. 0\Rightarrow\frac{\partial B_{y}}{\partial y} \right.} \\{= {- \frac{\partial B_{z}}{\partial z}}}\end{matrix} & \;\end{matrix}$

With these relationships among the derivatives the expressions for themagnetic vector potential terms can be simplified:

$\begin{matrix}\begin{matrix}{\frac{\partial B_{z}}{\partial y} = \left. \frac{\partial B_{y}}{\partial z}\Rightarrow{\overset{\rightarrow}{A}}_{1} \right.} \\{= {{\frac{1}{2} \cdot \frac{\partial B_{z}}{\partial y}}\left( {z^{2} - \left( {y - y_{0}} \right)^{2}} \right){\hat{u}}_{x}}}\end{matrix} & \; \\\begin{matrix}{\frac{\partial^{2}B_{z}}{\partial y^{2}} = {\frac{\partial\;}{\partial y}\frac{\partial B_{z}}{\partial y}}} \\{= {\frac{\partial\;}{\partial y}\frac{\partial B_{y}}{\partial z}}} \\{= {\frac{\partial\;}{\partial z}\frac{\partial B_{y}}{\partial y}}} \\{= {{- \frac{\partial\;}{\partial z}}\frac{\partial B_{z}}{\partial z}}} \\{= \left. {- \frac{\partial^{2}B_{z}}{\partial z^{2}}}\rightarrow{\overset{\rightarrow}{A}}_{2} \right.} \\{= {\frac{1}{3!}\frac{\partial^{2}B_{z}}{\partial y^{2}}\left( {y - y_{0}} \right)\left( {{3z^{2}} - \left( {y - {y\; 02{ux}}} \right.} \right.}}\end{matrix} & \; \\\begin{matrix}{\frac{\partial^{3}B_{z}}{{\partial y}{\partial z^{2}}} = {\frac{\partial\;}{\partial y}\frac{\partial^{2}B_{z}}{\partial z^{2}}}} \\{= {\frac{\partial\;}{\partial y}\left( {- \frac{\partial^{2}B_{z}}{\partial y^{2}}} \right)}} \\{{= {- \frac{\partial^{3}B_{z}}{\partial y^{3}}}};}\end{matrix} & \; \\\begin{matrix}{\frac{\partial^{3}B_{y}}{\partial z^{3}} = {\frac{\partial^{2}\;}{\partial z^{2}}\frac{\partial B_{y}}{\partial z}}} \\{= {\frac{\partial^{2}\;}{\partial z^{2}}\frac{\partial B_{z}}{\partial y}}} \\{= \frac{\partial^{3}B_{z}}{\partial y^{3}}} \\{= \left. {- \frac{\partial^{3}B_{z}}{\partial y^{3}}}\rightarrow{\overset{\rightarrow}{A}}_{3} \right.} \\{= {{\frac{1}{4!} \cdot \frac{\partial^{3}B_{z}}{\partial y^{3}}}\left( {{6\left( {y - y_{0}} \right)^{2}z^{2}} - z^{4} - \left( {y - y_{0}} \right)^{4}} \right){\hat{u}}_{x}}}\end{matrix} & \; \\\begin{matrix}{\frac{\partial^{4}B_{z}}{{\partial y^{2}}{\partial z^{2}}} = {\frac{\partial^{2}\;}{\partial y^{2}}\frac{\partial^{2}B_{z}}{\partial z^{2}}}} \\{{= {- \frac{\partial^{4}B_{z}}{\partial y^{4}}}};}\end{matrix} & \; \\\begin{matrix}{\frac{\partial^{4}B_{z}}{\partial z^{4}} = {\frac{\partial^{2}}{\partial z^{2}}\frac{\partial^{2}B_{z}}{\partial z^{2}}}} \\{= {{- \frac{\partial^{2}}{\partial z^{2}}}\frac{\partial^{2}B_{z}}{\partial y^{2}}}} \\{= {- \frac{\partial^{4}B_{z}}{{\partial y^{2}}{\partial z^{2}}}}} \\{= \left. \frac{\partial^{4}B_{z}}{\partial y^{4}}\rightarrow \right.}\end{matrix} & \;\end{matrix}$

gives the expression

${\overset{\rightarrow}{A}}_{4} = {{\frac{1}{5!} \cdot \frac{\partial^{4}B_{z}}{\partial y^{4}}}\left( {y - y_{0}} \right)\left( {{10\left( {y - y_{0}} \right)^{2}z^{2\;}} - {5z^{4}} - \left( {y - y_{0}} \right)^{4}} \right){\hat{u}}_{x}}$

The magnetic vector potential terms are therefore:

$\begin{matrix}{{\overset{\rightarrow}{A}}_{0} = {{- {B_{z}^{0}\left( {y - y_{0}} \right)}}{\hat{u}}_{x}}} & (27) \\{{\overset{\rightarrow}{A}}_{1} = {{\frac{1}{2} \cdot \frac{\partial B_{z}}{\partial y}}\left( {z^{2} - \left( {y - y_{0}} \right)^{2}} \right){\hat{u}}_{x}}} & (28) \\{{\overset{\rightarrow}{A}}_{2} = {{\frac{1}{3!} \cdot \frac{\partial^{2}B_{z}}{\partial y^{2}}}\left( {y - y_{0}} \right)\left( {{3z^{2}} - \left( {y - y_{0}} \right)^{2}} \right){\hat{u}}_{x}}} & (29) \\{{\overset{\rightarrow}{A}}_{3} = {{\frac{1}{4!} \cdot \frac{\partial^{3}B_{z}}{\partial y^{3}}}\left( {{6\left( {y - y_{0}} \right)^{2}z^{2}} - z^{4} - \left( {y - y_{0}} \right)^{4}} \right){\hat{u}}_{x}}} & (30) \\{{\overset{\rightarrow}{A}}_{4} = {{\frac{1}{5!} \cdot \frac{\partial^{4}B_{z}}{\partial y^{4}}}\left( {y - y_{0}} \right)\left( {{10\left( {y - y_{0}} \right)^{2}z^{2}} - {5z^{4}} - \left( {y - y_{0}} \right)^{4}} \right){\hat{u}}_{x}}} & (31)\end{matrix}$

The ideal field, given by the vector potential {right arrow over (A)}₀,is responsible for the ideal motion of a single particle in the trap,governed by

$H_{0} = {\frac{\left( {\overset{\rightarrow}{p} - {\overset{\rightarrow}{A}}_{0}} \right)^{2}}{2m} + {q \cdot {{\varphi \left( {{x;y},z} \right)}.}}}$

The terms {right arrow over (A)}₁, {right arrow over (A)}₂, {right arrowover (A)}₃, {right arrow over (A)}₄ . . . , represent deviations to theideal magnetic field, and produce perturbations of the ideal motion. Themain effect of these perturbations is the change of the trappingfrequencies ω_(p), ω_(z) and ω_(m). If the deviations, {right arrow over(A)}₁, {right arrow over (A)}₂, {right arrow over (A)}₃, {right arrowover (A)}₄ . . . , are not eliminated, then the eigen-frequencies becomedependent on the energies of the trapped particle. The cyclotron,magnetron and axial energies are denoted by E_(p), E_(m) and E_(z),respectively. The variation of the frequencies with the energies can becalculated using classical canonical perturbation theory (see forinstance the book by H. Goldstein “Classical Mechanics”). As an example,the effect of {right arrow over (A)}₂ is calculated below.

The term {right arrow over (A)}₂ represents a magnetic bottle, since itis created by the curvature of the magnetic field

$\frac{\partial^{2}B_{z}}{\partial y^{2}}.$

Remembering that the perturbation Hamiltonian is

${{\Delta \; H} = {{{- \frac{q}{m}}{\overset{\rightarrow}{p} \cdot \Delta}\; \overset{\rightarrow}{A}} + {\frac{q^{2}}{m}{{\overset{\rightarrow}{A}}_{0} \cdot \Delta}\; \overset{\rightarrow}{A}}}},$

where ΔA must be substituted by {right arrow over (A)}₂. Performing thecorresponding algebra (taking into account that {right arrow over(p)}=m{right arrow over (ν)}+q{right arrow over (A)}₀), the expressionfor the corresponding perturbative Hamiltonian

${\Delta \; {H(t)}} = {{{- \frac{q}{6}} \cdot \frac{\partial^{2}B_{z}}{\partial y^{2}}}{\overset{.}{x}\left( {y - y_{0}} \right)}\left( {{3z^{2}} - \left( {y - y_{0}} \right)^{2}} \right)}$

is obtained. With this, first order perturbation theory can be appliedto obtain the frequency shifts matrix (equivalent to the frequencyshifts matrix for electric anharmonicities introduced in equation 10).Proceeding as in that case, the matrix of the frequency deviationscaused by the magnetic bottle is the following:

$\begin{matrix}{\begin{pmatrix}{\Delta \; v_{p}} \\{\Delta \; v_{z}} \\{\Delta \; v_{m}}\end{pmatrix} = {\frac{{qB}_{2}}{8\pi^{3}m^{2}}\begin{pmatrix}\frac{\eta_{p}^{3}}{\gamma_{p}^{2}v_{p}^{2}} & {- \frac{\eta_{p}\xi_{p}}{\gamma_{p}v_{z}^{2}}} & \frac{\eta_{p}{\eta_{m}\left( {{\eta_{p}\xi_{m}v_{m}} + {\eta_{m}\xi_{p}v_{p}}} \right)}}{\gamma_{p}{v_{p}\left( {v_{m}^{2} - \frac{v_{z}^{2}}{2}} \right)}} \\{- \frac{\eta_{p}\xi_{p}}{\gamma_{p}v_{z}v_{p}}} & 0 & {- \frac{\eta_{m}\xi_{m}v_{m}}{v_{z}\left( {v_{m}^{2} - \frac{v_{z}^{2}}{2}} \right)}} \\\frac{\eta_{p}\eta_{m}{v_{m}\left( {{\eta_{p}\xi_{m}v_{m}} + {\eta_{m}\xi_{p}v_{p}}} \right)}}{\gamma_{p}{v_{p}^{2}\left( {v_{m}^{2} - \frac{v_{z}^{2}}{2}} \right)}} & {- \frac{\eta_{m}\xi_{m}v_{m}^{2}}{v_{z}^{2}\left( {v_{m}^{2} - \frac{v_{z}^{2}}{2}} \right)}} & \frac{\eta_{m}^{3}\xi_{m}v_{m}^{2}}{\left( {v_{m}^{2} - \frac{v_{z}^{2}}{2}} \right)}\end{pmatrix}\begin{pmatrix}{\Delta \; E_{p}} \\{\Delta \; E_{z}} \\{\Delta \; E_{m}}\end{pmatrix}}} & (32)\end{matrix}$

In equation (32) the inhomogeneity of the magnetic bottle has beenintroduced

$B_{2} = {\frac{1}{2!}{\frac{\partial^{2}B_{z}}{\partial y^{2}}.}}$

In general, the magnetic inhomogeneities are defined as

$B_{n} = {\frac{1}{n!}{\frac{\partial^{n}B_{z}}{\partial y^{n}}.}}$

Expressions equivalent to those of equation (32) can be derived for anyB_(n). The expressions are not further relevant; important is the factthat if not eliminated the magnetic inhomogeneities produce fluctuationsin the frequencies of the trapped charged particles with the energies.These fluctuations would render the technology not useful for massspectrometry, circuit-QED or any of the applications envisaged. Now,according to equations 27-31, the terms B₁, B₂, B₃, B₄ . . . B_(n) fullydefine the overall homogeneity of the magnetic field in the trappingposition (0, y₀, 0). The essential idea is that our technology providesthe means to eliminate all inhomogeneities B₁, B₂, B₃, B₄ . . . B_(n).This magnetic compensation is achieved with the magnetic elementsincluded in the chip, that is, with the so-called shim-pairs. In FIG. 1the magnetic elements 15,16 and 17,18 allow for eliminating thecoefficients B₁ and B₂. In general, the quantity of inhomogeneities thatcan be eliminated n is equal to the number n of shim-pairs in the chip.

Compensation of magnetic inhomogeneities B₁, B₂, B₃, B₄ . . . B_(n) bythe chip is illustrated below. For the sake of mathematical simplicityit will be assumed that the magnetic field created by the magneticelements is well described with the formula for a magnetic field createdby an infinitely long and thin wire (μ₀ is the magnetic permeability):

$\begin{matrix}{\overset{\rightarrow}{B} = {\frac{\mu_{0}I}{2\pi \sqrt{y^{2} + \left( {d/2} \right)^{2}}}\left( {{\frac{d/2}{\sqrt{y^{2} + \left( {d/2} \right)^{2}}}{\hat{u}}_{z}} + {\frac{y}{\sqrt{y^{2} + \left( {d/2} \right)^{2}}}{\hat{u}}_{y}}} \right)\quad}} & (33)\end{matrix}$

Equation 33 is valid for a wire placed at d/2 to the right of the mainwire (element 14 in FIG. 1). If the finite rectangular cross section orthe finite length of the wire is to be taken into account, then themathematical formula in equation (33) will be different, but in any casethe magnetic field will be proportional to the current I running alongthe wire (or the current density J). For a wire placed at d/2 to theleft of the main wire (element 14 in FIG. 1) the magnetic field is(again assuming an infinitely long and thin wire)

$\begin{matrix}{\overset{\rightarrow}{B} = {\frac{\mu_{0}I}{2\pi \sqrt{y^{2} + \left( {d/2} \right)^{2}}}\left( {{\frac{d/2}{\sqrt{y^{2} + \left( {d/2} \right)^{2}}}{\hat{u}}_{z}} - {\frac{y}{\sqrt{y^{2} + \left( {d/2} \right)^{2}}}{\hat{u}}_{y}}} \right)}} & (34)\end{matrix}$

With equations 33 and 34, the magnetic field created by one shim pair atthe symmetry axis of the chip (û_(y)) is given. The total field of theshim-pair is therefore:

$\begin{matrix}{\overset{->}{B} = {\frac{\mu_{0}{Id}}{2{\pi \left( {y^{2} + \left( {d\text{/}2} \right)^{2}} \right)}}{\hat{u}}_{z}}} & (35)\end{matrix}$

With equation 35 the derivatives of the magnetic field created by theshim-pair at the vertical axes (0, y, 0) can be easily obtained. Theformulas are:

$\begin{matrix}{{{\frac{\partial B_{z}}{\partial y} = {{- \frac{\mu_{0}I}{2\pi}}\frac{2\mspace{11mu} d\mspace{11mu} y}{\left( {y^{2} + \left( \frac{d}{2} \right)^{2}} \right)^{2}}}};}{\frac{\partial^{2}B_{z}}{\partial y^{2}} = {{- \frac{\mu_{0}I}{2\pi}} \cdot \frac{d\left( {d^{2} - {12y^{2}}} \right)}{\left( {y^{2} + \left( \frac{d}{2} \right)^{2}} \right)^{3}}}}{{\frac{\partial^{3}B_{z}}{\partial y^{2}} = {\frac{\mu_{0}I}{2\pi} \cdot \frac{d\mspace{11mu} y\mspace{11mu} \left( {d^{2} - {4y^{2}}} \right)}{6\left( {y^{2} + \left( \frac{d}{2} \right)^{2}} \right)^{4}}}};}{\frac{\partial^{4}B_{z}}{\partial y^{4}} = {\frac{\mu_{0}I}{2\pi} \cdot \frac{3d\; \left( {d^{4} - {40d^{2}y^{2}} + {80y^{4}}} \right)}{2\left( {y^{2} + \left( {d\text{/}2} \right)^{2}} \right)^{5}}}}} & (36)\end{matrix}$

The magnetic compensation (elimination of B₁, B₂, B₃, B₄ . . . B_(n))can now be illustrated with the formulas of equation (36). It is assumedthat the chip is fabricated with 4 such shim-pairs plus the main wire.Each shim-pair is placed at a distance d_(i)/2 from the main wire, withthe current I_(i) and the corresponding magnetic field denoted by B_(z)^(i). The compensation means finding the currents of the shim-pairs I₁,I₂, I₃ and I₄ such that the total inhomogeneity up to the fourth ordervanishes:

$\begin{matrix}{{{\frac{\partial B_{z}^{1}}{\partial y} + \frac{\partial B_{z}^{2}}{\partial y} + \frac{\partial B_{z}^{3}}{\partial y} + \frac{\partial B_{z}^{4}}{\partial y}} = {- \frac{\partial B_{z}^{0}}{\partial y}}}{{\frac{\partial^{2}B_{z}^{1}}{\partial y^{2}} + \frac{\partial^{2}B_{z}^{2}}{\partial y^{2}} + \frac{\partial^{2}B_{z}^{3}}{\partial y^{2}} + \frac{\partial^{2}B_{z}^{4}}{\partial y^{2}}} = {- \frac{\partial^{2}B_{z}^{0}}{\partial y^{2}}}}{{\frac{\partial^{3}B_{z}^{1}}{\partial y^{3}} + \frac{\partial^{3}B_{z}^{2}}{\partial y^{3}} + \frac{\partial^{3}B_{z}^{3}}{\partial y^{3}} + \frac{\partial^{3}B_{z}^{4}}{\partial y^{3}}} = {- \frac{\partial^{3}B_{z}^{0}}{\partial y^{3}}}}{{\frac{\partial^{4}B_{z}^{1}}{\partial y^{4}} + \frac{\partial^{4}B_{z}^{2}}{\partial y^{4}} + \frac{\partial^{4}B_{z}^{3}}{\partial y^{4}} + \frac{\partial^{4}B_{z}^{4}}{\partial y^{4}}} = {- \frac{\partial^{4}B_{z}^{0}}{\partial y^{4}}}}} & (37)\end{matrix}$

The compensation in equation 37 shows that the currents I_(i) have toadjusted such that the inhomogeneity of the main wire

$\frac{\partial^{n}B_{z}^{0}}{\partial y^{n}}$

is compensated by the inhomogeneities of the shim-pairs. With theexpressions for the derivatives of equation (36) this gives:

$\begin{matrix}{{{{- \frac{\mu_{0}I_{1}}{2\pi}}\frac{2\mspace{11mu} d_{1}\; y}{\left( {y^{2} + \left( \frac{d_{1}}{2} \right)^{2}} \right)^{2}}} - {\frac{\mu_{0}I_{2}}{2\pi}\frac{2\mspace{11mu} d_{2}\; y}{\left( {y^{2} + \left( \frac{d_{2}}{2} \right)^{2}} \right)^{2}}} - {\frac{\mu_{0}I_{3}}{2\pi}\frac{2\mspace{11mu} d_{3}\; y}{\left( {y^{2} + \left( \frac{d_{3}}{2} \right)^{2}} \right)^{2}}} - {\frac{\mu_{0}I_{4}}{2\pi}\frac{2\mspace{11mu} d_{4}\; y}{\left( {y^{2} + \left( \frac{d_{4}}{2} \right)^{2}} \right)^{2}}}} = {{{- \frac{\partial B_{z}^{0}}{\partial y}} - {\frac{\mu_{0}I_{1}}{2\pi} \cdot \frac{d_{1}\left( {d_{1}^{2} - {12y^{2}}} \right)}{\left( {y^{2} + \left( \frac{d_{1}}{2} \right)^{2}} \right)^{3}}} - {\frac{\mu_{0}I_{2}}{2\pi} \cdot \frac{d_{2}\left( {d_{2}^{2} - {12y^{2}}} \right)}{\left( {y^{2} + \left( \frac{d_{2}}{2} \right)^{2}} \right)^{3}}} - {\frac{\mu_{0}I_{3}}{2\pi} \cdot \frac{d_{3}\left( {d_{3}^{2} - {12y^{2}}} \right)}{\left( {y^{2} + \left( \frac{d_{3}}{2} \right)^{2}} \right)^{3}}} - {\frac{\mu_{0}I_{4}}{2\pi} \cdot \frac{d_{4}\left( {d_{4}^{2} - {12y^{2}}} \right)}{\left( {y^{2} + \left( \frac{d_{4}}{2} \right)^{2}} \right)^{3}}}} = {{{{- \frac{\partial^{2}B_{z}^{0}}{\partial y^{2}}}{\frac{\mu_{0}I_{1}}{2\pi} \cdot \frac{d_{1}\; y\; \left( {d_{1}^{2} - {4y^{2}}} \right)}{6\left( {y^{2} + \left( \frac{d_{1}}{2} \right)^{2}} \right)^{4}}}} + {\frac{\mu_{0}I_{2}}{2\pi} \cdot \frac{d_{2}\mspace{11mu} y\; \left( {d_{2}^{2} - {4y^{2}}} \right)}{6\left( {y^{2} + \left( \frac{d_{2}}{2} \right)^{2}} \right)^{4}}} + {\frac{\mu_{0}I_{3}}{2\pi} \cdot \frac{d_{3}\mspace{11mu} y\mspace{11mu} \left( {d_{3}^{2} - {4y^{2}}} \right)}{6\left( {y^{2} + \left( \frac{d_{3}}{2} \right)^{2}} \right)^{4}}} + {\frac{\mu_{0}I_{4}}{2\pi} \cdot \frac{d_{4}\; y\mspace{11mu} \left( {d_{4}^{2} - {4y^{2}}} \right)}{6\left( {y^{2} + \left( \frac{d_{4}}{2} \right)^{2}} \right)^{4}}}} = {{{{- \frac{\partial^{3}B_{z}^{0}}{\partial y^{3}}}{\frac{\mu_{0}I_{1}}{2\pi} \cdot \frac{3d_{1}\left( {d_{1}^{4} - {40d_{1}^{2}y^{2}} + {80y^{4}}} \right)}{2\left( {y^{2} + \left( \frac{d_{1}}{2} \right)^{2}} \right)^{5}}}} + {\frac{\mu_{0}I_{2}}{2\pi} \cdot \frac{3d_{2}\left( {d_{2}^{4} - {40d_{2}^{2}y^{2}} + {80y^{4}}} \right)}{2\left( {y^{2} + \left( \frac{d_{2}}{2} \right)^{2}} \right)^{5}}} + {\frac{\mu_{0}I_{3}}{2\pi} \cdot \frac{3d_{3}\left( {d_{3}^{4} - {40d_{3}^{2}y^{2}} + {80y^{4}}} \right)}{2\left( {y^{2} + \left( \frac{d_{3}}{2} \right)^{2}} \right)^{5}}} + {\frac{\mu_{0}I_{4}}{2\pi} \cdot \frac{3d_{4}\left( {d_{4}^{4} - {40d_{4}^{2}y^{2}} + {80y^{4}}} \right)}{2\left( {y^{2} + \left( \frac{d_{4}}{2} \right)^{2}} \right)^{5}}}} = {- \frac{\partial^{4}B_{z}^{0}}{\partial y^{4}}}}}}} & (38)\end{matrix}$

Equations (38) can be expressed in Matrix form as follows:

$\begin{matrix}{{\begin{pmatrix}\overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} \\\frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} \\\frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} \\\frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)}\end{pmatrix}\begin{pmatrix}\; \\\;\end{pmatrix}} = {\frac{\pi}{\mu}\begin{pmatrix}\frac{\partial}{\partial} \\\frac{\partial}{\partial} \\{- \frac{\partial}{\partial}} \\{- \frac{\partial}{\partial}}\end{pmatrix}}} & (39)\end{matrix}$

The magnetic Matrix of the chip (for the case n=4) is given by:

$\begin{matrix}{= \begin{pmatrix}\overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} & \overset{\_}{\left( {+ (/)} \right)} \\\frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} \\\frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} & \frac{( - )}{\left( {+ (/)} \right)} \\\frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- \mspace{11mu} +} \right)}{\left( {+ (/)} \right)}\end{pmatrix}} & (40)\end{matrix}$

In the expression of the matrix MB, the general position coordinate ymust be substituted by the position of the trapped particles y₀ when thecompensation is to be performed at that position (which is usually thecase). In general the determinant of MB is different form zero,therefore the matrix MB can be inverted. With the inverted matrix MB⁻¹the proper compensation currents to be applied to the shim-pairs can beeasily found:

$\begin{matrix}{\begin{pmatrix}I_{1} \\\vdots \\I_{n}\end{pmatrix} = {\frac{2\pi}{\mu_{0}}{{MB}^{- 1}\begin{pmatrix}{- \frac{\partial B_{z}^{0}}{\partial y}} \\\vdots \\{- \frac{\partial^{n}B_{z}^{0}}{\partial y^{n}}}\end{pmatrix}}}} & (41)\end{matrix}$

In general the design of a concrete chip consists in optimizing thematrix MB, such that the positions of the shim-pairs relative to themain wire (d_(i)) will be chosen in order to guarantee that the valuesof the currents obtained from equation (41) are below the value for thecritical current of the superconducting material employed. Theexpression of MB in equation (40) has been derived assuming infinitelylong and infinitely thin wires. If the wires are not infinitely thin,the expression for MB will be different, but the generality of thesolution (41) is still valid, since the linearity of the problem offinding the compensation currents is guaranteed by the universal lineardependence of the magnetic field with the current I (or current densityJ) resulting from Maxwell's equations. The shim-pairs must bedistributed symmetrically (along the û_(z) axis) with respect to themain wire (14 in FIG. 1). The shim-pairs 15,16 and 17,18 shown in FIG. 1are placed on the same plane as the main wire 14, but this is notstrictly necessary. The shim-pairs 15, 16, 17, 18 may be placed in planeabove the main wire, and it may be also possible to place differentshim-pairs at different planes. In general the shim-pairs 15, 16, 17, 18are responsible for cancelling the magnetic inhomogeneities but not forcreating the bulk magnetic field. Therefore it is mostly convenient thatthe cross section of the shim wires be smaller than that of the mainwire. The decision of placing the shim-wires whether at the same planeas the main wire or at a different plane will also depend on the costsof fabrication of the chip.

In general the design of a particular chip is determined by the matrixMB. The thickness of the shim-wires, the positions d and the planes atwhich they are placed can vary, but the condition sine qua non for agood chip design is that the matrix MB must be invertible at y₀ and thesolutions for the shim-currents of equation 41 must be “physical”, i.e.the values of I₁ . . . I_(n) must be sustainable by the materialsemployed.

To illustrate the general working principle with a concrete example, thefollowing values for the dimensions of the magnetic elements areassumed:

l_(p) = 50.00 mm w_(p) = 10.00 mm (main magnetic element) l_(s1) = 50/15mm w_(s1) ⁼ w_(p)/3 mm d_(s1) = 10.0 mm (1^(st) shim-pair) l_(s2) =50/15 mm w_(s2) ⁼ w_(p)/3 mm d_(s2) = 20.0 mm (2^(nd) shim-pair) l_(s3)= 50/15 mm w_(s3) ⁼ w_(p)/3 mm d_(s3) = 30.0 mm (3^(rd) shim-pair)l_(s4) = 50/10 mm w_(s4) = w_(p)/3 mm d_(s4) = 45.0 mm (4^(th)shim-pair)

It is assumed the magnetic elements implementing the shim-pairs 31, 32,33, 34 are all placed on top of the main magnetic element 14. This isshown in FIG. 19. The trapped electron is assumed to be at a positiony₀=0.820 mm above the surface of the array of electrodes (not shown inFIG. 19). Hence, the magnetic elements 14, 31, 32, 33, 34 plotted inFIG. 19 substitute the array 13 of magnetic elements 14, 15, 16, 17, 18shown in FIG. 1. A current density along the main magnetic element 14 ofJ₀=81.48 A/mm² is assumed. The main magnetic element 14 may be a solidblock of superconducting material properly magnetized to sustain anequivalent current density. Alternatively the main magnetic element 14can be made of an array of thin superconducting wires. In the lattercase, for instance, assuming thin wires with a radius of 0.25/2 mm, itwould be necessary to have an array of 200 turns distributed over 40vertical layers which in total would occupy the same volume as the solidblock of cross section l_(p)×w_(p). The current running long each of thethin wires would be 4.0 Ampere. In both cases, the wires or the solidblock must be long “enough”, such that at the position of the trappedion (0, y₀, 0) only current components along the direction û_(y) arevisible. The ideal solution would be, as mentioned previously, to haveinfinitely long wires. Two practical solutions circumvent this demand.They will be discussed in detail in later paragraphs, now for thepurpose of this illustrating example, it is simply assumed that thewires are infinitely long. The matrix MB for the example is:

$\begin{matrix}{{MB} = {\begin{pmatrix}0.874002 & 0.369749 & 0.181921 & 0.127718 \\{- 0.502579} & {- 0.0540262} & {- 0.0119514} & {- 0.00380739} \\0.0565019 & {- 0.0133297} & {- 0.00394869} & {- 0.00140489} \\0.169422 & 0.00852214 & 0.000957463 & 0.000146981\end{pmatrix}\mspace{14mu} 1\text{/}{mm}^{2}}} & (42)\end{matrix}$

The inhomogeneities of the magnetic field produced by the main magneticelement are given by (units Gauss/mm^(n)):

$\begin{matrix}{\begin{pmatrix}\frac{\partial B_{z}^{0}}{\partial y} \\\frac{\partial^{2}B_{z}^{0}}{\partial y^{2}} \\\frac{\partial^{3}B_{z}^{0}}{\partial y^{3}} \\\frac{\partial^{4}B_{z}^{0}}{\partial y^{4}}\end{pmatrix} = \begin{pmatrix}{- \frac{114.29}{1!}} \\\frac{2.814}{2!} \\\frac{0.17455}{3!} \\{- \frac{0.03453}{4!}}\end{pmatrix}} & (43)\end{matrix}$

Solving for the currents in equation 41 gives the shim-currentdensities:

$\begin{matrix}{\begin{pmatrix}J_{1} \\J_{2} \\J_{3} \\J_{4}\end{pmatrix} = {\begin{pmatrix}11.708 \\{- 429.699} \\1885.17 \\{- 626.481}\end{pmatrix}\mspace{14mu} A\text{/}{mm}^{2}}} & (44)\end{matrix}$

All current densities are below the critical values for Niobium Titaniumor YBCO (below 2 T fields). The example also shows that in order toavoid the current densities becoming too high, it is convenient to havethe outer shim-pairs with a bigger cross section than the inner ones.This can be easily achieved, simply by increasing l_(sn) and/or w_(sn).The decision on the values for those parameters will depend on theenvisaged values for magnetic fields at the planned range for y₀, aswell as possible fabrication issues. Thus, different shim-pairs mayconveniently have different l_(sn), w_(sn).

FIG. 20 shows the magnetic field (û_(z) component) along the z-axes forboth the main magnetic element 14 alone and for the main magneticelement 14 plus the four shim-pairs 31, 32, 33, 34. The graph iscalculated at the position of the trapped particles y₀. It must benoticed that the amplitude of the motion of the trapped particles alongû_(z) is below 1 mm for cryogenic temperatures (typically 30 μm for oneelectron at 4.2 K). Therefore, the relevant range within which the totalmagnetic field must be homogeneous is centred around z=0 (see FIG. 20)and spans only a few mm. The symmetric peaks of the magnetic field atz≅15 mm are due to the third shim-pair 33, but their position is too faraway from z=0 to affect the motion of the trapped particles.

FIG. 21 shows the magnetic field (û_(y) component) along the z-axes forboth the main magnetic element 14 alone and for the main magneticelement 14 plus the four shim-pairs 31, 32, 33, 34. The graph shows thatthanks to the magnetic compensation, the û_(y) component of the totalmagnetic field disappears. The remaining magnetic field is purely axial,i.e. with only a û_(z) component. This means the ion trap may workprecisely. A component of the field along the ûl_(y) axes would producebig shifts of the eigen-frequencies ω_(p), ω_(z) and ω_(m) with theparticles' energies. This is avoided with the magnetic compensation. Itmust be noticed in FIG. 21, that the range within which B_(y)≅0 spans afew mm, much wider than the actual amplitude of the particles' motion atcryogenic temperatures. Hence, within the relevant range of a few mmaround (0, y₀, 0) the compensated magnetic field is oriented alongû_(z).

FIG. 22 shows the magnetic field (û_(z) component) along the z-axes forboth, the main magnetic element alone and for the main magnetic elementplus the four shim-pairs. The graph is a zoom of FIG. 20 around (0, y₀,0). It shows in detail the effect of the magnetic compensation along thez-axes.

FIG. 23 shows the magnetic field (û_(z) component) along the y-axes forboth, the main magnetic element 14 alone and for the main magneticelement 14 plus the four shim-pairs 31, 32, 33, 34. The positive effectof the magnetic compensation is clearly visible. While the magneticfield of the main magnetic element 14 alone drops rapidly with thevertical distance (y) to the surface of the chip, the compensatedmagnetic field shows a flat plateau centred around the position of thetrapped charged particles (0,y₀, 0). The flat plateau spans a few mm, inany case much bigger than the amplitude of the motion of the particlesat cryogenic temperatures. This shows that the compensation properlyforms a homogenous magnetic field region. This is the key factor, whichrenders the technology useful for ion trapping and applications incircuit-QED, high-precision mass spectrometry and others.

Now it is necessary to address the question regarding the finite lengthof the employed magnetic elements. Below are provided two possibletechnical solutions to this issue. The magnetic elements can be a)either closed loop wires carrying a superconducting current density orb) magnetised blocks of superconducting material with a constant andhomogeneous magnetic dipole moment density. In case a) the trappingregion must be enclosed by a superconducting shielding case, such thatfrom the position (0, y₀,0) the magnetic elements (14,15,16,17,18 inFIGS. 2 and 14, 31, 32, 33, 34 in FIG. 19) only currents running alongthe axis 14. FIGS. 24 and 25 show a sketch of the chip enclosed by asuperconducting shielding box 35 (the electrodes of the ion trap insidethe box 35 are not shown). The example shows how the magnetic elements14, 31 are actually closed loop wires, along which a persistentsuperconducting current is running. With the superconducting shield, thetrapping region around (0, y₀, 0) only “sees” currents running along theaxis û_(x). Therefore, the hypotheses of the mathematic analysisdescribed in the preceding paragraphs are perfectly met: the assumedsymmetries do apply and the results obtained are fully valid. If thesuperconducting case, or any other type of magnetic shielding, is notincluded then the closed loop wires will have current components runningin directions other than û_(x) and the validity of the exposedmathematical analysis is not exact. This means that the degree ofhomogeneity of the magnetic field achieved at (0, y₀,0) may be lowerthan that achieved with the shielding case. The proposed technologywould still work, permitting all the applications listed in thepreceding sections, however ultra high precision will not be possible atthe same level as it is with the use of the magnetic shield.

Case b) is now referred to where the magnetic elements are magnetisedblocks of superconducting material with a constant and homogeneousmagnetic dipole moment density. In this case, the structure of themagnetic elements is simplified over the case a), and the use of asuperconducting shielding case is not required to avoid currentcomponents in directions other than û_(x). A sketch of the magneticelements 14, 31 is shown in FIG. 26. In that figure M₀ is the magneticdipole density of the main magnetic element and M₁ is the magneticdipole density of the first shim-pair. Both M₀ and M₁ have to behomogenous densities. This type of magnetic structures can be fabricatedpreferentially with high temperature superconducting materials, such asYBCO, but also with any other ferromagnetic materials, such as Cobalt,Iron and Nickel.

FIG. 26 does not show the elements required for magnetizing the magneticelements. There are different known ways of magnetising/demagnetisinghigh temperature superconductors or ferromagnets. For instance by usingthermal pulses applied to the superconductors in the presence of a small“seeding” magnetic field. The magnetisation methods have been describedelsewhere and are known to the skilled person. The same argument appliesto the case of using closed loop low temperature superconductors (seeFIGS. 24 and 25). These closed-loop wires must be energized/de energizedin situ, that is, during the operation of the ion trap. Again, differenttechnologies are available, such as superconducting flux pumps orroom-temperature power supplies employed with cryogenic superconductingswitches. One of these technologies will be necessarily part of the iontrap; the employed option may be different, depending on the particularapplication envisaged for a particular ion trap. All these energizingtechnologies have been described in detail elsewhere and are known tothe skilled person.

The magnetic field of the earth may be taken into account for a correctcompensation of the total magnetic field in the trapping region. Now,the Earth's magnetic field is of the order of 0.5 Gauss depending on theposition. For the scale of the motion of the trapped charged particlesin the ion trap (several microns), the earth's field, {right arrow over(B)}_(earth), can be considered as homogeneous. However, in general itwill have components in all directions of space:

{right arrow over (B)} _(earth)=(B _(x) ^(earth) ,B _(y) ^(earth) ,B_(z) ^(earth))  (45)

With the ion trap 1 as shown in FIGS. 1 and 2 and the magnetic elementsas shown in FIGS. 19, 24, 25 and 26 it is possible to compensate for they component of the earth's field: B_(y) ^(earth)=0. The component alongthe z-axes, B_(z) ^(earth), will simply add to the total trapping fieldalong that direction, thus it has no negative effect upon the operationof the ion trap. The component along the x-axes however, does affectnegatively the operation of the trap. With the magnetic elements shownin FIGS. 1, 2, 19, 24, 25 and 26, B_(x) ^(earth) cannot be eliminated.The earth's magnetic field may be eliminated within the experimentalregion with some external coils. However, the optimum solution is toemploy similar magnetic compensation shim-pairs as those shown in thementioned figures, however rotated 90 degrees along the y-axes. Theresulting total magnetic structure is sketched in FIG. 27 (with theelectrodes of the ion trap 1, above the magnetic elements, not shown).The magnetic elements 36, 37 are now oriented parallel to the z-axes andso is the sense of the compensation currents I₀ ^(x), I₁ ^(x) . . .I_(m) ^(x). Figure shows 27 only one shim-pair 37 for compensating B_(x)^(earth) (or any remaining magnetic field component along the x-axes,whatever its origin). More shim-pairs could be added, up to m, dependingon the degree of compensation to required. The shim-pairs 37 are placedsymmetrically across the x-axes. There is one main magnetic element 36,with current I₀ ^(x), and up to m shim-pairs with currents I₁ ^(x) . . .I_(m) ^(x). The shim pairs 37 are symmetrical with respect to the mainmagnetic element for compensating B_(x) ^(earth). The principle of thecompensation is identical to what has been discussed in the previoussections. In this case however the shim-currents (denoted by have to bechosen such that:

$\begin{matrix}{\begin{pmatrix}I_{1}^{x} \\\vdots \\I_{m}^{x}\end{pmatrix} = {\frac{2\pi}{\mu_{0}}{{MB}_{x}^{- 1}\begin{pmatrix}{- \frac{\partial\left( {B_{x}^{earth} + B_{x}^{0}} \right)}{\partial y}} \\\vdots \\{- \frac{\partial^{n}\left( {B_{x}^{earth} + B_{x}^{0}} \right)}{\partial y^{n}}}\end{pmatrix}}}} & (46)\end{matrix}$

In equation (46) the matrix MB_(x) is similar to the matrix MB ofequation 40, however for the shim-pairs employed for the compensation ofB_(x) ^(earth) (see FIG. 27). Equation (46) states that theshim-currents have to be chosen such, to eliminate the totalinhomogeneities of the earth's magnetic field plus those due to the mainmagnetic element 36 for the compensation of B_(x) (see FIG. 27). Thefield created by that main magnetic element 36 is here denoted by B_(x)⁰. Now, the main current I₀ ^(x) has to be chosen such that the earth'smagnetic field vanishes at the trapping position (0, y₀, 0). Thecondition to be fulfilled by I₀ ^(x) is therefore: B_(x) ^(earth)+B_(x)⁰+(B_(x) ¹+ . . . +B_(x) ^(m))=0. This condition takes also into accountthe effect of the compensation currents upon the total x-component ofthe magnetic field: the total field along x must vanish. In equation 46the earth's field can be considered homogeneous, so the equation can besimplified to:

$\begin{matrix}{\begin{pmatrix}I_{1}^{x} \\\vdots \\I_{m}^{x}\end{pmatrix} \cong {\frac{2\pi}{\mu_{0}}{{MB}_{x}^{- 1}\begin{pmatrix}{- \frac{\partial B_{x}^{0}}{\partial y}} \\\vdots \\{- \frac{\partial^{n}B_{x}^{0}}{\partial y^{n}}}\end{pmatrix}}}} & (47)\end{matrix}$

The current I₀ ^(x) along the main magnetic element 36 for thecompensation of B_(x) ^(earth) is shown in FIG. 27. It must be observedthat I₀ ^(x) flows along the û_(z)-axes. It is also important to observethat the compensation currents I₁ ^(x) . . . I_(m) ^(x) also flow alongthe û_(z)-axes. Additionally, every I_(i) ^(x) propagates in the samesense for both components of the pair, as shown in FIG. 27.

To finalize this section it is necessary to consider now thecompensation of the component B_(y) ^(earth) (or in general they-component of the magnetic field whatever its origin may be). Thecompensation of B_(y) ^(earth) is achieved with the additionalcompensation currents I_(y) ⁰, I_(y) ¹, . . . I_(y) ^(q). In this case,the main compensation current is I_(y) ⁰ and it runs along a pair ofwires and along the û_(z)-axes. This is shown in FIG. 28 (electrodes ofthe ion trap 1, above the magnetic elements, are not shown and magneticelements for compensating B_(x), shown in FIG. 27, not shown). Themagnetic elements 38, 39 (wires or superconducting ferromagnets) areparallel to the magnetic elements 36, 37 employed for compensating B_(x)^(earth). Each magnetic pair 38, 39 is placed symmetrically across theû_(x)-axes (i.e. there is invariance under the transformation x→-x).This is sketched in FIG. 29. Moreover, it is important to observe thatthe current I_(y) ⁰ runs in opposite senses in each component of thepair. This is necessary for the magnetic field created by that pair tobe oriented along the û_(y)-axes at the trapping position (0, y₀, 0).The main compensation current I_(y) ⁰ must be chosen such, that theoverall component of the magnetic field along the y-axes vanishes: B_(y)^(earth)+B_(y) ⁰+(B_(y) ¹+ . . . +B_(y) ^(q))=0. The shim currents areI_(y) ¹, . . . I_(y) ^(q), they are required to eliminate theinhomogeneities of the magnetic field created by I_(y) ⁰ plus themagnetic field of the earth along the vertical (y) axes. These currentsmust be chosen such that:

$\begin{matrix}{\begin{pmatrix}I_{1}^{y} \\\vdots \\I_{q}^{y}\end{pmatrix} = {\frac{2\pi}{\mu_{0}}{{MB}_{y}^{- 1}\begin{pmatrix}{- \frac{\partial\left( {B_{y}^{earth} + B_{y}^{0}} \right)}{\partial y}} \\\vdots \\{- \frac{\partial^{n}\left( {B_{y}^{earth} + B_{y}^{0}} \right)}{\partial y^{n}}}\end{pmatrix}}}} & (48)\end{matrix}$

Assuming infinitely long thin wires, the magnetic Matrix MB_(y) of thechip (for the case q=4) is given by:

$\begin{matrix}{= \begin{pmatrix}{- \; \frac{- (/)}{\left( {+ (/)} \right)}} & {- \; \frac{- (/)}{\left( {+ (/)} \right)}} & {- \; \frac{- (/)}{\left( {+ (/)} \right)}} & {- \; \frac{- (/)}{\left( {+ (/)} \right)}} \\\frac{\left( {- (/)} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- (/)} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- (/)} \right)}{\left( {+ (/)} \right)} & \frac{\left( {- (/)} \right)}{\left( {+ (/)} \right)} \\{- \; \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}} & {- \; \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}} & {- \; \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}} & {- \; \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}} \\{\; \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}} & \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)} & \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)} & \frac{\left( {{- \; —} + (/)} \right)}{\left( {+ (/)} \right)}\end{pmatrix}} & (49)\end{matrix}$

In equation 49, d_(i) is the distance of the corresponding shim-wire tothe plane x=0 (see FIGS. 28 and 29). It must be noticed that, forsimplicity, those figures show only the main compensation pair 38 (forthe current I₀ ^(y)) and one shim-pair 39 (for the current I₁ ^(y)).This is the minimal configuration for compensating the magnetic field ofthe earth along the y-axes. Similarly, for the compensation of B_(x)^(earth), the minimal configuration required is the main compensationelement 36 for the current I₀ ^(x) plus the first shim-pair 37 for thecurrent I₁ ^(x), as shown in FIG. 27.

The magnetic elements for the compensation of the earth's magnetic fieldcan be fabricated with low or high temperature superconductors. Thedecision depends on the particular application. The field to becompensated is very small: B_(x) ^(earth)≅B_(y) ^(earth)≦1 Gauss,therefore no high currents are required. This means that the crosssection of the magnetic elements can be very small, of the order of 1mm² or less. For ease of fabrication, the cross section of all elementsshould rather be the same. However, eventually different shim-pairs mayalso have different cross sections. The separation between the differentmagnetic elements is preferably also of that order of magnitude, around1 mm or lower, although higher separations would also work.

FIG. 30 shows a complete ion chip, including the electrodes for creatingthe trapping electrostatic potential, the magnetic elements for creatingthe trapping magnetic field and the magnetic elements for eliminatingthe earth's magnetic field along the û_(x) axes, B_(x) ^(earth) andalong the û_(y) axes, B_(y) ^(earth).

The distribution of the currents can be different to that shown in theFIGS. 28,29. For instance, what in FIG. 29 is shown to sustain thecurrent I₂ ^(x) may be used instead to sustain I₀ ^(y) and vice versa.Since the fabrication of the structure does not exclude any of thesedifferent options, the optimal current distribution will be decided bythe user, simply by applying the desired current to the desired magneticelement in whatever order is most convenient for him. There is only oneexception to this, namely the main magnetic element for compensatingB_(x) ^(earth) (see FIG. 27), which can only be used for that purpose:it cannot be used for compensating B_(y) ^(earth). This magnetic elementmust be parallel to the û_(z)-axes, and its symmetry axes shouldcoincide with the plane x=0. The rest of the magnetic elements for thecompensation of B_(x) ^(earth) and B_(y) ^(earth) must also be orientedalong the û_(z)-axes and symmetrically distributed across the plane x=0.

As for the case of the upper magnetic structure for generating thehomogenous trapping magnetic field along the z-axes, the superconductingcurrents or magnetic dipole densities may be applied with techniquesdescribed elsewhere as known to the skilled person. It is also necessarythat, as seen from the trapping position (0, y₀, 0) the magneticelements for compensating the earth's magnetic field show only currentspropagating along û_(z), with no components along the axis û_(x), û_(y).This is achieved with the use of the superconducting shielding case, assketched in FIG. 25.

Other variations and modifications will be apparent to the skilledperson. Such variations and modifications may involve equivalent andother features which are already known and which may be used instead of,or in addition to, features described herein. Features that aredescribed in the context of separate embodiments may be provided incombination in a single embodiment. Conversely, features which aredescribed in the context of a single embodiment may also be providedseparately or in any suitable sub-combination.

It should be noted that the term “comprising” does not exclude otherelements or steps, the term “a” or “an” does not exclude a plurality, asingle feature may fulfil the functions of several features recited inthe claims and reference signs in the claims shall not be construed aslimiting the scope of the claims. It should also be noted that theFigures are not necessarily to scale; emphasis instead generally beingplaced upon illustrating the principles of the present invention.

1. An ion trap comprising: a first array of magnetic elements arrangedto generate a first magnetic field with a degree of homogeneity; and anarray of electrodes arranged to generate an electrostatic fieldincluding a turning point in electrical potential at a location wherethe magnetic field has a substantially maximum degree of homogeneity;wherein the array of electrodes is planar and parallel to the directionof the magnetic field at the location; and wherein a primary firstmagnetic element is arranged to generate a first component of the firstmagnetic field and other first magnetic elements are arranged togenerate compensating components of the first magnetic field that reducethe gradient, the curvature and higher order derivatives of the firstcomponent of the first magnetic field at the location where the firstmagnetic field has the substantially maximum degree of homogeneity. 2.The ion trap of claim 1, wherein the electrodes of the array each havesurfaces facing the location where the magnetic field is substantiallyhomogeneous, which surfaces are substantially coplanar.
 3. The ion trapof claim 1, wherein the array of electrodes comprises a row of three ormore electrodes, which row is arranged to be parallel to the directionof the magnetic field at the location where the magnetic field issubstantially homogeneous.
 4. The ion trap of claim 3, wherein the rowcomprises five electrodes.
 5. The ion trap of claim 3, wherein thelengths of the electrodes along the direction of the row are such thatan electrode in the middle of the row is shortest and electrodes at theends of the row are longest.
 6. The ion trap of claim 3, comprising aguard electrode on each side of the row.
 7. The ion trap of claim 6,wherein the guard electrodes overlap the electrodes of the row.
 8. Theion trap of claim 1, wherein the array of electrodes is provided on asubstrate and the array of magnetic elements is provided on the samesubstrate.
 9. The ion trap of claim 1, wherein the array of magneticelements comprises a row of magnetic elements, which row extends in thesame direction as the row of electrodes.
 10. The ion trap of claim 1,wherein the magnetic elements each comprise a wire arranged to conductan electric current.
 11. The ion trap of claim 1, wherein the electrodearray is provided on a top surface of the substrate and the array ofmagnetic elements is provided below the electrode array.
 12. The iontrap preceding claim 1, wherein the degree of homogeneity of themagnetic field is predetermined.
 13. The ion trap of claim 12, whereinthe predetermined degree of homogeneity of the magnetic field isobtained by empirically adjusting the magnetic field of the array ofmagnetic elements.
 14. The ion trap of claim 1, wherein the other firstmagnetic elements comprise at least one pair of first magnetic elements,the elements within each at least one pair being arranged to havesubstantially equal currents running therethrough in the same direction.15. The ion trap of claim 1, further comprising a second array ofmagnetic elements arranged to compensate for a first component of anexternal magnetic field.
 16. The ion trap of claim 15, wherein themagnetic elements of the second array are substantially perpendicular tothe magnetic elements of the first array.
 17. The ion trap of claim 15,wherein a primary second magnetic element is arranged to generate afirst component of a second magnetic field and other second magneticelements are arranged to generate compensating components of the secondmagnetic field that reduce the gradient and curvature of the firstcomponent of the second magnetic field and compensate for the firstcomponent external magnetic field at the location where the firstmagnetic field has the substantially maximum degree of homogeneity. 18.The ion trap according to claim 17, wherein the other second magneticelements comprise at least one pair of second magnetic elements, theelements within each at least one pair being arranged to havesubstantially equal currents running therethrough in the same direction.19. The ion trap of claim 1, further comprising a third array ofmagnetic elements arranged to compensate for a second component of anexternal magnetic field.
 20. The ion trap of claim 19, wherein themagnetic elements of the third array are substantially perpendicular tothe magnetic elements of the first array.
 21. The ion trap of claim 19,wherein third magnetic elements are arranged to generate a thirdmagnetic field to compensate for a second component of the externalmagnetic field at the location where the first magnetic field has thesubstantially maximum degree of homogeneity.
 22. The ion trap accordingto claim 21, wherein the third magnetic elements comprise at least onepair of third magnetic element, the elements within each at least onepair being arranged to have substantially equal currents runningtherethrough in mutually opposite directions.
 23. The ion trap of claim15, wherein the external magnetic field is the earth's magnetic field.24. A mass spectrometer comprising the ion trap of claim
 1. 25. Amicrowave quantum circuit comprising the ion trap of claim
 1. 26. Amethod of trapping an ion, the method comprising: using an array ofmagnetic elements to generate a magnetic field with a degree ofhomogeneity; and using an array of electrodes to generate anelectrostatic field including a turning point in electrical potential ata location where the magnetic field has a maximum degree of homogeneity;wherein the array of electrodes is planar and parallel to the directionof the magnetic field at the location; and wherein a primary magneticelement is arranged to generate a first component of the magnetic fieldand other magnetic elements are arranged to generate compensatingcomponents of the magnetic field that reduce the gradient, the curvatureand higher order derivatives of the first component of the magneticfield at the location where the magnetic field has the substantiallymaximum degree of homogeneity.
 27. The method of claim 26, wherein thedegree of homogeneity of the magnetic field is predetermined.
 28. Themethod of claim 27, further comprising obtaining the predetermineddegree of homogeneity of the magnetic field by empirically adjusting themagnetic field of the array of magnetic elements.
 29. The method ofclaim 28, wherein empirically adjusting the magnetic field of the arrayof magnetic elements comprises probing the magnetic field with amagnetic sensor.
 30. The method of claim 29, wherein the magnetic sensoris the ion to be trapped.